Statistical Distributions in Simulation R.B. Lenin - PowerPoint PPT Presentation

Statistical Distributions in Simulation R.B. Lenin (rblenin@daiict.ac.in) Autumn 2007 R.B. Lenin (rblenin@daiict.ac.in) () Statistical Distributions in Simulation Autumn 2007 1 / 68

Outline Random Variables 1 Introduction Discrete random variables Continuous random variable Mean of a random variable Properties of µ Variance of a random variable Properties of σ 2 Covariance of random variables Correlation of random variables Properties of σ 2 Correlation between random variables Chebyshev’s inequality Strong law of large numbers Discrete Distributions 2 Bernoulli random variable Binomial random variable Geometric random variable Poisson random variable Poisson approximation Poisson process Continuous Distributions 3 Uniform random variable Exponential random variable Erlang random variable Hyper-exponential random variable Gamma random variable Weibull random variable Beta random variable Triangular random variable Normal (Gaussian) random variable Central limit theorem (CLT) Log-normal random variable R.B. Lenin (rblenin@daiict.ac.in) () Statistical Distributions in Simulation Autumn 2007 2 / 68

Random Variables Introduction Random variables Discrete lottery coin tossing number of customers in a supermarket R.B. Lenin (rblenin@daiict.ac.in) () Statistical Distributions in Simulation Autumn 2007 3 / 68

Random Variables Introduction Random variables Discrete lottery coin tossing number of customers in a supermarket Continuous interarrival times of vehicles at a traffic light water level in a dam time taken for a chemical reaction R.B. Lenin (rblenin@daiict.ac.in) () Statistical Distributions in Simulation Autumn 2007 3 / 68

Random Variables Discrete random variables Discrete random variables A random variable X is said to be discrete if it can take on at most a countable number of values, say, x 1 , x 2 , . . . , . The probability that X is equal to x i is given by p ( x i ) = Pr { X = x i } , i = 1 , 2 , . . . and ∞ � p ( x i ) = 1 . i =1 R.B. Lenin (rblenin@daiict.ac.in) () Statistical Distributions in Simulation Autumn 2007 4 / 68

❘ Random Variables Discrete random variables Discrete random variables · · · p ( x ) is known as the probability mass function (pmf). R.B. Lenin (rblenin@daiict.ac.in) () Statistical Distributions in Simulation Autumn 2007 5 / 68

Random Variables Discrete random variables Discrete random variables · · · p ( x ) is known as the probability mass function (pmf). The cumulative distribution function (cdf) F ( x ) is given by � F X ( x ) = Pr { X ≤ x } = p ( x i ) , ∀ x ∈ ❘ . x i ≤ x R.B. Lenin (rblenin@daiict.ac.in) () Statistical Distributions in Simulation Autumn 2007 5 / 68

Random Variables Continuous random variable Continuous random variable A random variable X is said to be continuous if there exists a nonnegative function f ( x ), the probability density function , such that for any set of real numbers B , � Pr { X ∈ B} = f ( x ) dx B and � ∞ f ( x ) dx = 1 . −∞ R.B. Lenin (rblenin@daiict.ac.in) () Statistical Distributions in Simulation Autumn 2007 6 / 68

Random Variables Continuous random variable Continuous random variable · · · If x is a number and ∆ x > 0, then � x +∆ x Pr { X ∈ [ x , x + ∆ x ] } = f ( y ) dy x and Pr { X = x } = 0 . R.B. Lenin (rblenin@daiict.ac.in) () Statistical Distributions in Simulation Autumn 2007 7 / 68

Random Variables Continuous random variable Continuous random variable · · · The cumulative distribution function F X ( x ) for a continuous random variable X is given by � x F X ( x ) = Pr { X ∈ ( −∞ , x ] } = f ( y ) dy , ∀ x ∈ ❘ . −∞ R.B. Lenin (rblenin@daiict.ac.in) () Statistical Distributions in Simulation Autumn 2007 8 / 68

Random Variables Mean of a random variable Mean of a random variable The mean or expected value of a random variable X , denoted by µ or E [ X ], is given by ∞  �  x i p ( x i ) , if X is discrete     i =1  E [ X ] = � ∞    xf ( x ) dx , if X is continuous    −∞ R.B. Lenin (rblenin@daiict.ac.in) () Statistical Distributions in Simulation Autumn 2007 9 / 68

Random Variables Properties of µ Properties of µ 1 E [ cX ] = cE [ X ], for any constant c . R.B. Lenin (rblenin@daiict.ac.in) () Statistical Distributions in Simulation Autumn 2007 10 / 68

Random Variables Properties of µ Properties of µ 1 E [ cX ] = cE [ X ], for any constant c . 2 E [ X + Y ] = E [ X ] + E [ Y ] regardless of whether X and Y are independent. R.B. Lenin (rblenin@daiict.ac.in) () Statistical Distributions in Simulation Autumn 2007 10 / 68

Random Variables Properties of µ Properties of µ 1 E [ cX ] = cE [ X ], for any constant c . 2 E [ X + Y ] = E [ X ] + E [ Y ] regardless of whether X and Y are independent. 3 E [ XY ] = E [ X ] E [ Y ], provided X and Y are independent. R.B. Lenin (rblenin@daiict.ac.in) () Statistical Distributions in Simulation Autumn 2007 10 / 68

Random Variables Properties of µ Properties of µ 1 E [ cX ] = cE [ X ], for any constant c . 2 E [ X + Y ] = E [ X ] + E [ Y ] regardless of whether X and Y are independent. 3 E [ XY ] = E [ X ] E [ Y ], provided X and Y are independent. The mean is one measure of the central tendency of a random variable. R.B. Lenin (rblenin@daiict.ac.in) () Statistical Distributions in Simulation Autumn 2007 10 / 68

Random Variables Variance of a random variable Variance of a random variable The variance of the random variable X , denoted by σ 2 or Var [ X ], is given by σ 2 = E [( X − µ ) 2 ] = E [ X 2 ] − µ 2 R.B. Lenin (rblenin@daiict.ac.in) () Statistical Distributions in Simulation Autumn 2007 11 / 68

Random Variables Variance of a random variable Variance of a random variable The variance of the random variable X , denoted by σ 2 or Var [ X ], is given by σ 2 = E [( X − µ ) 2 ] = E [ X 2 ] − µ 2 The variance is a measure of the dispersion of a random variable about its mean. R.B. Lenin (rblenin@daiict.ac.in) () Statistical Distributions in Simulation Autumn 2007 11 / 68

Properties of σ 2 Random Variables Properties of σ 2 1 Var [ cX ] = c 2 Var [ X ] R.B. Lenin (rblenin@daiict.ac.in) () Statistical Distributions in Simulation Autumn 2007 12 / 68

Properties of σ 2 Random Variables Properties of σ 2 1 Var [ cX ] = c 2 Var [ X ] 2 Var [ X + Y ] = Var [ X ] + Var [ Y ] , if X and Y are independent. R.B. Lenin (rblenin@daiict.ac.in) () Statistical Distributions in Simulation Autumn 2007 12 / 68

Properties of σ 2 Random Variables Properties of σ 2 1 Var [ cX ] = c 2 Var [ X ] 2 Var [ X + Y ] = Var [ X ] + Var [ Y ] , if X and Y are independent. The square root of the variance is called the standard deviation and is denoted by σ . R.B. Lenin (rblenin@daiict.ac.in) () Statistical Distributions in Simulation Autumn 2007 12 / 68

Properties of σ 2 Random Variables Properties of σ 2 1 Var [ cX ] = c 2 Var [ X ] 2 Var [ X + Y ] = Var [ X ] + Var [ Y ] , if X and Y are independent. The square root of the variance is called the standard deviation and is denoted by σ . It can be given the most definitive interpretation when X has a normal distribution. R.B. Lenin (rblenin@daiict.ac.in) () Statistical Distributions in Simulation Autumn 2007 12 / 68

Random Variables Covariance of random variables Covariance of random variables The covariance between the random variables X and Y , denoted by Cov [ X , Y ], is defined by Cov [ X , Y ] = E { [ X − E [ X ]][ Y − E [ Y ]] } = E [ XY ] − E [ X ][ Y ] R.B. Lenin (rblenin@daiict.ac.in) () Statistical Distributions in Simulation Autumn 2007 13 / 68

Random Variables Covariance of random variables Covariance of random variables The covariance between the random variables X and Y , denoted by Cov [ X , Y ], is defined by Cov [ X , Y ] = E { [ X − E [ X ]][ Y − E [ Y ]] } = E [ XY ] − E [ X ][ Y ] The covariance is a measure of the dependence between X and Y . R.B. Lenin (rblenin@daiict.ac.in) () Statistical Distributions in Simulation Autumn 2007 13 / 68

Random Variables Covariance of random variables Covariance of random variables The covariance between the random variables X and Y , denoted by Cov [ X , Y ], is defined by Cov [ X , Y ] = E { [ X − E [ X ]][ Y − E [ Y ]] } = E [ XY ] − E [ X ][ Y ] The covariance is a measure of the dependence between X and Y . Note that Cov [ X , X ] = Var [ X ]. R.B. Lenin (rblenin@daiict.ac.in) () Statistical Distributions in Simulation Autumn 2007 13 / 68

Random Variables Correlation of random variables Correlation of random variables Cov [ X , Y ] X and Y = 0 are uncorrelated > 0 are positively correlated < 0 are negatively correlated Independent random variables are uncorrelated R.B. Lenin (rblenin@daiict.ac.in) () Statistical Distributions in Simulation Autumn 2007 14 / 68

Properties of σ 2 Random Variables Properties of σ 2 Var [ X − Y ] = Var [ X ] + Var [ Y ] − 2 Cov [ X , Y ] R.B. Lenin (rblenin@daiict.ac.in) () Statistical Distributions in Simulation Autumn 2007 15 / 68

Properties of σ 2 Random Variables Properties of σ 2 Var [ X − Y ] = Var [ X ] + Var [ Y ] − 2 Cov [ X , Y ] If X and Y are independent, then Var [ X − Y ] = Var [ X ] + Var [ Y ] R.B. Lenin (rblenin@daiict.ac.in) () Statistical Distributions in Simulation Autumn 2007 15 / 68