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Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals. s s s s s s ✲ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals. s s s s s s ✲ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals. s s s s s s ✲ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals. s s s s s s ✲ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals. s s s s s s ✲ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals. s s s s s s ✲ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals. s s s s s s ✲ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals. s s s s s s ✲ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals. s s s s s s ✲ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals. s s s s s s ✲ x 0 = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals. s s s s s s ✲ x 0 = 0 x n = T logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals. s s s s s s ✲ x 1 x 0 = 0 x n = T logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals. s s s s s s ✲ x 1 x 2 x 0 = 0 x n = T logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals. s s s s s s ✲ x 1 x 2 x 3 x 0 = 0 x n = T logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals. s s s s s s ✲ x 1 x 2 x 3 x 4 x 0 = 0 x n = T logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals. s s s s s s ✲ x 1 x 2 x 3 x 4 x 5 x 0 = 0 x n = T logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals. s s s s s s ✲ x 1 x 2 x 3 x 4 x 5 x 6 x 0 = 0 x n = T logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals. s s s s s s ✲ x 1 x 2 x 3 x 4 x 5 x 6 x 0 = 0 x n = T ··· logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals. s s s s s s ✲ x 1 x 2 x 3 x 4 x 5 x 6 x n − 1 x 0 = 0 x n = T ··· logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals. s s s s s s ✲ x 1 x 2 x 3 x 4 x 5 x 6 x n − 1 x 0 = 0 x n = T ··· That means there is a λ > 0 so that the probability that an event occurs in a given time interval is p = λ n . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals. s s s s s s ✲ x 1 x 2 x 3 x 4 x 5 x 6 x n − 1 x 0 = 0 x n = T ··· That means there is a λ > 0 so that the probability that an event occurs in a given time interval is p = λ n . We keep this λ independent of the number of subintervals. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals. s s s s s s ✲ x 1 x 2 x 3 x 4 x 5 x 6 x n − 1 x 0 = 0 x n = T ··· That means there is a λ > 0 so that the probability that an event occurs in a given time interval is p = λ n . We keep this λ independent of the number of subintervals. That way, if we double the number of intervals, for each interval the probability of an event occurring is divided by 2. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals. s s s s s s ✲ x 1 x 2 x 3 x 4 x 5 x 6 x n − 1 x 0 = 0 x n = T ··· That means there is a λ > 0 so that the probability that an event occurs in a given time interval is p = λ n . We keep this λ independent of the number of subintervals. That way, if we double the number of intervals, for each interval the probability of an event occurring is divided by 2. These assumptions are realistic logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals. s s s s s s ✲ x 1 x 2 x 3 x 4 x 5 x 6 x n − 1 x 0 = 0 x n = T ··· That means there is a λ > 0 so that the probability that an event occurs in a given time interval is p = λ n . We keep this λ independent of the number of subintervals. That way, if we double the number of intervals, for each interval the probability of an event occurring is divided by 2. These assumptions are realistic for hits on web pages logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals. s s s s s s ✲ x 1 x 2 x 3 x 4 x 5 x 6 x n − 1 x 0 = 0 x n = T ··· That means there is a λ > 0 so that the probability that an event occurs in a given time interval is p = λ n . We keep this λ independent of the number of subintervals. That way, if we double the number of intervals, for each interval the probability of an event occurring is divided by 2. These assumptions are realistic for hits on web pages, for customer support calls received logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Take the time interval and cut it up into n subintervals. Assume that an event is equally likely to occur in any of the subintervals. s s s s s s ✲ x 1 x 2 x 3 x 4 x 5 x 6 x n − 1 x 0 = 0 x n = T ··· That means there is a λ > 0 so that the probability that an event occurs in a given time interval is p = λ n . We keep this λ independent of the number of subintervals. That way, if we double the number of intervals, for each interval the probability of an event occurring is divided by 2. These assumptions are realistic for hits on web pages, for customer support calls received, for radioactive particles registered. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process We will also assume that no two events can happen in the same interval. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process We will also assume that no two events can happen in the same interval. For a sufficiently small time scale (large n ) this is realistic in the above examples. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process We will also assume that no two events can happen in the same interval. For a sufficiently small time scale (large n ) this is realistic in the above examples. s s s s s s ✲ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process We will also assume that no two events can happen in the same interval. For a sufficiently small time scale (large n ) this is realistic in the above examples. s s s s s s ✲ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process We will also assume that no two events can happen in the same interval. For a sufficiently small time scale (large n ) this is realistic in the above examples. s s s s s s ✲ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process We will also assume that no two events can happen in the same interval. For a sufficiently small time scale (large n ) this is realistic in the above examples. s s s s s s ✲ x 0 = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process We will also assume that no two events can happen in the same interval. For a sufficiently small time scale (large n ) this is realistic in the above examples. s s s s s s ✲ x 0 = 0 x 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process We will also assume that no two events can happen in the same interval. For a sufficiently small time scale (large n ) this is realistic in the above examples. s s s s s s ✲ x 0 = 0 x 1 x 2 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process We will also assume that no two events can happen in the same interval. For a sufficiently small time scale (large n ) this is realistic in the above examples. s s s s s s ✲ x 0 = 0 x 1 x 2 x 3 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process We will also assume that no two events can happen in the same interval. For a sufficiently small time scale (large n ) this is realistic in the above examples. s s s s s s ✲ x 0 = 0 x 1 x 2 x 3 x 4 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process We will also assume that no two events can happen in the same interval. For a sufficiently small time scale (large n ) this is realistic in the above examples. s s s s s s ✲ x 0 = 0 x 1 x 2 x 3 x 4 x 5 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process We will also assume that no two events can happen in the same interval. For a sufficiently small time scale (large n ) this is realistic in the above examples. s s s s s s ✲ x 0 = 0 x 1 x 2 x 3 x 4 x 5 x 6 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process We will also assume that no two events can happen in the same interval. For a sufficiently small time scale (large n ) this is realistic in the above examples. s s s s s s ✲ x 0 = 0 x 1 x 2 x 3 x 4 x 5 x 6 ··· logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process We will also assume that no two events can happen in the same interval. For a sufficiently small time scale (large n ) this is realistic in the above examples. s s s s s s ✲ x n − 1 x 0 = 0 x 1 x 2 x 3 x 4 x 5 x 6 ··· logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process We will also assume that no two events can happen in the same interval. For a sufficiently small time scale (large n ) this is realistic in the above examples. s s s s s s ✲ x n − 1 x 0 = 0 x 1 x 2 x 3 x 4 x 5 x 6 x n = T ··· logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process We will also assume that no two events can happen in the same interval. For a sufficiently small time scale (large n ) this is realistic in the above examples. s s s s s s ✲ x n − 1 x 0 = 0 x 1 x 2 x 3 x 4 x 5 x 6 x n = T ··· Plus, we could argue that within time intervals of a certain length only one event can be registered. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Under the assumptions that logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Under the assumptions that the interval we investigate has n subintervals logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Under the assumptions that the interval we investigate has n subintervals so that the probability an event occurs in a subinterval is p = λ n logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Under the assumptions that the interval we investigate has n subintervals so that the probability an event occurs in a subinterval is p = λ n and that at most one event occurs in each subinterval, logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Under the assumptions that the interval we investigate has n subintervals so that the probability an event occurs in a subinterval is p = λ n and that at most one event occurs in each subinterval, the probability that x events occur overall is given by the binomial distribution. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Under the assumptions that the interval we investigate has n subintervals so that the probability an event occurs in a subinterval is p = λ n and that at most one event occurs in each subinterval, the probability that x events occur overall is given by the binomial distribution. b ( x ; n , p ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Under the assumptions that the interval we investigate has n subintervals so that the probability an event occurs in a subinterval is p = λ n and that at most one event occurs in each subinterval, the probability that x events occur overall is given by the binomial distribution. � n � p x ( 1 − p ) n − x b ( x ; n , p ) = x logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Under the assumptions that the interval we investigate has n subintervals so that the probability an event occurs in a subinterval is p = λ n and that at most one event occurs in each subinterval, the probability that x events occur overall is given by the binomial distribution. � n � n ! p x ( 1 − p ) n − x = b ( x ; n , p ) = x ( n − x ) ! x ! logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Under the assumptions that the interval we investigate has n subintervals so that the probability an event occurs in a subinterval is p = λ n and that at most one event occurs in each subinterval, the probability that x events occur overall is given by the binomial distribution. λ x � n � n ! p x ( 1 − p ) n − x = b ( x ; n , p ) = n x x ( n − x ) ! x ! logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Under the assumptions that the interval we investigate has n subintervals so that the probability an event occurs in a subinterval is p = λ n and that at most one event occurs in each subinterval, the probability that x events occur overall is given by the binomial distribution. � n − x λ x � n � � n ! 1 − λ p x ( 1 − p ) n − x = b ( x ; n , p ) = n x x ( n − x ) ! x ! n logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Under the assumptions that the interval we investigate has n subintervals so that the probability an event occurs in a subinterval is p = λ n and that at most one event occurs in each subinterval, the probability that x events occur overall is given by the binomial distribution. � n − x λ x � n � � n ! 1 − λ p x ( 1 − p ) n − x = b ( x ; n , p ) = n x x ( n − x ) ! x ! n � n � n − x λ x �� 1 − λ n ! n = ( n − x ) ! n x x ! n logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Under the assumptions that the interval we investigate has n subintervals so that the probability an event occurs in a subinterval is p = λ n and that at most one event occurs in each subinterval, the probability that x events occur overall is given by the binomial distribution. � n − x λ x � n � � n ! 1 − λ p x ( 1 − p ) n − x = b ( x ; n , p ) = n x x ( n − x ) ! x ! n � n � n − x λ x �� 1 − λ n ! n = ( n − x ) ! n x x ! n n → ∞ 1 − → logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Under the assumptions that the interval we investigate has n subintervals so that the probability an event occurs in a subinterval is p = λ n and that at most one event occurs in each subinterval, the probability that x events occur overall is given by the binomial distribution. � n − x λ x � n � � n ! 1 − λ p x ( 1 − p ) n − x = b ( x ; n , p ) = n x x ( n − x ) ! x ! n � n � n − x λ x �� 1 − λ n ! n = ( n − x ) ! n x x ! n 1 · λ x n → ∞ − → x ! logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Under the assumptions that the interval we investigate has n subintervals so that the probability an event occurs in a subinterval is p = λ n and that at most one event occurs in each subinterval, the probability that x events occur overall is given by the binomial distribution. � n − x λ x � n � � n ! 1 − λ p x ( 1 − p ) n − x = b ( x ; n , p ) = n x x ( n − x ) ! x ! n � n � n − x λ x �� 1 − λ n ! n = ( n − x ) ! n x x ! n 1 · λ x n → ∞ x ! · e − λ − → logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Under the assumptions that the interval we investigate has n subintervals so that the probability an event occurs in a subinterval is p = λ n and that at most one event occurs in each subinterval, the probability that x events occur overall is given by the binomial distribution. � n − x λ x � n � � n ! 1 − λ p x ( 1 − p ) n − x = b ( x ; n , p ) = n x x ( n − x ) ! x ! n � n � n − x λ x �� 1 − λ n ! n = ( n − x ) ! n x x ! n 1 · λ x n → ∞ x ! · e − λ − → (We let n → ∞ because cutting up the original time interval was just a way to get the model.) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Definition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Definition. A random variable is said to have a Poisson distribution with parameter λ > 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Definition. A random variable is said to have a Poisson distribution with parameter λ > 0 if and only if its probability mass function is p ( x ; λ ) = e − λ λ x x ! logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Definition. A random variable is said to have a Poisson distribution with parameter λ > 0 if and only if its probability mass function is p ( x ; λ ) = e − λ λ x x ! This really is a probability mass function, because of the series representation of the exponential function logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Definition. A random variable is said to have a Poisson distribution with parameter λ > 0 if and only if its probability mass function is p ( x ; λ ) = e − λ λ x x ! This really is a probability mass function, because of the series ∞ λ x representation of the exponential function: e λ = ∑ x ! . x = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Example. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Example. The number of calls a certain customer support department receives in any given 5 minute interval is Poisson distributed with λ = 2 . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Example. The number of calls a certain customer support department receives in any given 5 minute interval is Poisson distributed with λ = 2 . What is the probability that at most 3 calls are received in the next 5 minutes? logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Example. The number of calls a certain customer support department receives in any given 5 minute interval is Poisson distributed with λ = 2 . What is the probability that at most 3 calls are received in the next 5 minutes? p ( 0;2 ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Example. The number of calls a certain customer support department receives in any given 5 minute interval is Poisson distributed with λ = 2 . What is the probability that at most 3 calls are received in the next 5 minutes? p ( 0;2 )+ p ( 1;2 ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Example. The number of calls a certain customer support department receives in any given 5 minute interval is Poisson distributed with λ = 2 . What is the probability that at most 3 calls are received in the next 5 minutes? p ( 0;2 )+ p ( 1;2 )+ p ( 2;2 ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Example. The number of calls a certain customer support department receives in any given 5 minute interval is Poisson distributed with λ = 2 . What is the probability that at most 3 calls are received in the next 5 minutes? p ( 0;2 )+ p ( 1;2 )+ p ( 2;2 )+ p ( 3;2 ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Example. The number of calls a certain customer support department receives in any given 5 minute interval is Poisson distributed with λ = 2 . What is the probability that at most 3 calls are received in the next 5 minutes? p ( 0;2 )+ p ( 1;2 )+ p ( 2;2 )+ p ( 3;2 ) = 0 . 857 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process Example. The number of calls a certain customer support department receives in any given 5 minute interval is Poisson distributed with λ = 2 . What is the probability that at most 3 calls are received in the next 5 minutes? p ( 0;2 )+ p ( 1;2 )+ p ( 2;2 )+ p ( 3;2 ) = 0 . 857 (Used a Poisson distribution table to look up p ( x ≤ 3;2 ) , which is just a value of the cumulative distribution function.) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process For large values of n , the Poisson distribution can be used to approximate binomial probabilities. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process For large values of n , the Poisson distribution can be used to approximate binomial probabilities. Example. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process For large values of n , the Poisson distribution can be used to approximate binomial probabilities. Example. Suppose the probability of a missing page in an individual book a certain manufacturer makes is 0 . 1 . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process For large values of n , the Poisson distribution can be used to approximate binomial probabilities. Example. Suppose the probability of a missing page in an individual book a certain manufacturer makes is 0 . 1 . What is the probability that a batch of 200 books has at most 10 books with a page missing? logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process For large values of n , the Poisson distribution can be used to approximate binomial probabilities. Example. Suppose the probability of a missing page in an individual book a certain manufacturer makes is 0 . 1 . What is the probability that a batch of 200 books has at most 10 books with a page missing? Using a Poisson distribution, we have that logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process For large values of n , the Poisson distribution can be used to approximate binomial probabilities. Example. Suppose the probability of a missing page in an individual book a certain manufacturer makes is 0 . 1 . What is the probability that a batch of 200 books has at most 10 books with a page missing? Using a Poisson distribution, we have that λ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process For large values of n , the Poisson distribution can be used to approximate binomial probabilities. Example. Suppose the probability of a missing page in an individual book a certain manufacturer makes is 0 . 1 . What is the probability that a batch of 200 books has at most 10 books with a page missing? Using a Poisson distribution, we have that λ = np logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process For large values of n , the Poisson distribution can be used to approximate binomial probabilities. Example. Suppose the probability of a missing page in an individual book a certain manufacturer makes is 0 . 1 . What is the probability that a batch of 200 books has at most 10 books with a page missing? Using a Poisson distribution, we have that λ = np = 200 · 0 . 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution

Motivation Derivation Definition Examples Expected Value/Variance The Poisson Process For large values of n , the Poisson distribution can be used to approximate binomial probabilities. Example. Suppose the probability of a missing page in an individual book a certain manufacturer makes is 0 . 1 . What is the probability that a batch of 200 books has at most 10 books with a page missing? Using a Poisson distribution, we have that λ = np = 200 · 0 . 1 = 20. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Poisson Distribution