Statistics for Business Random Variables and Probability - PowerPoint PPT Presentation

Statistics for Business Random Variables and Probability Distributions, Special Discrete and Continuous Probability Distributions Panagiotis Th. Konstantinou MSc in International Shipping, Finance and Management , Athens University of Economics and Business First Draft : July 15, 2045. This Draft : September 6, 2020. P. Konstantinou (AUEB) Statistics for Business – II September 6, 2020 1 / 67

Random Variables and Probability Distributions Random Variables: Intro Random Variables – I Basics Definition A random variable X is a a function or rule that assigns a number to each outcome of an experiment. Think of this as the numerical summary of a random outcome. Random Variables Discrete Continuous Random Variable Random Variable countable number of values any value in an interval P. Konstantinou (AUEB) Statistics for Business – II September 6, 2020 2 / 67

Random Variables and Probability Distributions Random Variables: Intro Random Variables – II Basics Examples X = GPA for a randomly selected student X = number of contracts a shipping company has pending at a randomly selected month of the year X = number on the upper face of a randomly tossed die X = the price of crude oil during a randomly selected month. P. Konstantinou (AUEB) Statistics for Business – II September 6, 2020 3 / 67

Random Variables and Probability Distributions Discrete Random Variables and Distributions Discrete Random Variables A discrete random variable can only take on a countable number of values Examples Roll a die twice. Let X be the number of times 4 comes up: ◮ then X could be 0 , 1 , or 2 times Toss a coin 5 times. Let X be the number of heads: ◮ then X = 0 , 1 , 2 , 3 , 4 , or 5 P. Konstantinou (AUEB) Statistics for Business – II September 6, 2020 4 / 67

Random Variables and Probability Distributions Discrete Random Variables and Distributions Discrete Probability Distributions – I The probability distribution for a discrete random variable X resembles the relative frequency distributions. It is a graph, table or formula that gives the possible values of X and the probability P ( X = x ) associated with each value. This must satisfy 0 ≤ P ( x ) ≤ 1 , for all x . 1 � all x P ( x ) = 1 , the individual probabilities sum to 1. 2 The cumulative probability function , denoted by F ( x 0 ) , shows the probability that X is less than or equal to a particular value, x 0 : � F ( x 0 ) = Pr( X ≤ x 0 ) = P ( x ) x ≤ x 0 P. Konstantinou (AUEB) Statistics for Business – II September 6, 2020 5 / 67

Random Variables and Probability Distributions Discrete Random Variables and Distributions Discrete Probability Distributions – II Random Experiment : Toss 2 Coins. Let (the random variable) X = # heads. Probability Distribution 4 possible outcomes T T x Value Probability Cum. Prob. 0 1/4 = .25 1/4 = .25 T H 1 2/4 = .50 3/4 = .75 2 1/4 = .25 4/4 = 1.00 H T .50 Probability H H .25 0 1 2 x P. Konstantinou (AUEB) Statistics for Business – II September 6, 2020 6 / 67

Random Variables and Probability Distributions Discrete Random Variables and Distributions Discrete Probability Distributions – III Random Experiment : Let the random variable S be the number of days it will snow in the last week of January (cumulative) Probability distribution of S Outcome 0 1 2 3 4 5 6 7 Probability 0.20 0.25 0.20 0.15 0.10 0.05 0.04 0.01 CDF 0.20 0.45 0.65 0.80 0.90 0.95 0.99 1.00 P. Konstantinou (AUEB) Statistics for Business – II September 6, 2020 7 / 67

Random Variables and Probability Distributions Discrete Random Variables and Distributions Moments of Discrete Prob. Distributions – I Expected Value (or mean ) of a discrete distribution ( weighted average ) � µ X = E ( X ) = x · P ( x ) . all x Variance of a discrete random variable X ( weighted average ...) � ( X − µ X ) 2 � � σ 2 = Var ( X ) = E ( x − µ X ) 2 · P ( x ) = all x Standard Deviation of a discrete random variable X �� √ σ 2 = σ = ( x − µ ) 2 P ( x ) all x P. Konstantinou (AUEB) Statistics for Business – II September 6, 2020 8 / 67

Random Variables and Probability Distributions Discrete Random Variables and Distributions Moments of Discrete Prob. Distributions – II Example Consider the experiment of tossing 2 coins, and X = # of heads. Then � µ = E ( X ) = x xP ( x ) = ( 0 × 0 . 25 ) + ( 1 × 0 . 50 ) + ( 2 × 0 . 25 ) = 1 �� σ = x ( x − µ ) 2 P ( x ) � = ( 0 − 1 ) 2 ( . 25 ) + ( 1 − 1 ) 2 ( . 50 ) + ( 2 − 1 ) 2 ( . 25 ) √ = . 50 = 0 . 707 P. Konstantinou (AUEB) Statistics for Business – II September 6, 2020 9 / 67

Random Variables and Probability Distributions Discrete Random Variables and Distributions Moments of Discrete Prob. Distributions – III Example (Number of days it will snow in January) µ S = E ( S ) = � s s · P ( s ) = = 0 · 0 . 2 + 1 · 0 . 25 + 2 · 0 . 2 + 3 · 0 . 15 + 4 · 0 . 1 + 5 · 0 . 05 + 6 · 0 . 04 + 7 · 0 . 01 = 2 . 06 s ( s − E ( S )) 2 · P ( s ) = S = Var ( S ) = � σ 2 = ( 0 − 2 . 06 ) 2 · 0 . 2 +( 1 − 2 . 06 ) 2 · 0 . 25 +( 2 − 2 . 06 ) 2 · 0 . 2 +( 3 − 2 . 06 ) 2 · 0 . 15 +( 4 − 2 . 06 ) 2 · 0 . 1 + ( 5 − 2 . 06 ) 2 · 0 . 05 + ( 6 − 2 . 06 ) 2 · 0 . 04 +( 7 − 2 . 06 ) 2 · 0 . 01 = 2 . 94 Remark (Rules for Moments) Let a and b be any constants and let Y = a + bX. Then E [ a + bX ] = a + b E [ X ] = a + b µ x b 2 Var [ X ] = b 2 σ 2 Var [ a + bX ] = x ⇒ σ Y = | b | σ x The above imply that E [ a ] = a and Var [ a ] = 0 P. Konstantinou (AUEB) Statistics for Business – II September 6, 2020 10 / 67

Random Variables and Probability Distributions Continuous Random Variables and Densities Prob. Density and Distribution Function – I The probability density function (or pdf ), f ( x ) , of continuous random variable X has the following properties f ( x ) > 0 for all values of x ( x takes a range of values, R X ). 1 The area under the probability density function f ( x ) over all 2 values of the random variable X is equal to 1 (recall that � all x P ( x ) = 1 for discrete r.v.) � f ( x ) dx = 1 . R X P. Konstantinou (AUEB) Statistics for Business – II September 6, 2020 11 / 67

Random Variables and Probability Distributions Continuous Random Variables and Densities Prob. Density and Distribution Function – II The probability that X lies between two values is the area under 3 the density function graph between the two values: � b Pr( a ≤ X ≤ b ) = Pr( a < X < b ) = f ( x ) dx a f ( x ) ≤ ≤ ( b ) P a x ) a = P ( < x < b a b x Note that the probability of any individual value is zero P. Konstantinou (AUEB) Statistics for Business – II September 6, 2020 12 / 67

Random Variables and Probability Distributions Continuous Random Variables and Densities Prob. Density and Distribution Function – III The cumulative density function (or distribution function ) F ( x 0 ) , 4 which expresses the probability that X does not exceed the value of x 0 , is the area under the probability density function f ( x ) from the minimum x value up to x 0 � x 0 F ( x 0 ) = f ( x ) dx . x min It follows that 5 Pr( a ≤ X ≤ b ) = Pr( a < X < b ) = F ( b ) − F ( a ) P. Konstantinou (AUEB) Statistics for Business – II September 6, 2020 13 / 67

Random Variables and Probability Distributions Continuous Random Variables and Densities Moments of Continuous Distributions – I Expected Value (or mean ) of a continuous distribution � µ X = E ( X ) = xf ( x ) dx . R X Variance of a continuous random variable X � σ 2 ( x − µ X ) 2 f ( x ) dx X = Var ( X ) = R X Standard Deviation of a continuous random variable X �� � σ X = σ 2 X = ( x − µ X ) 2 f ( x ) dx R X P. Konstantinou (AUEB) Statistics for Business – II September 6, 2020 14 / 67

Random Variables and Probability Distributions Continuous Random Variables and Densities Moments of Continuous Distributions – II Remark (Rules for Moments Apply) Let c and d be any constants and let Y = c + dX. Then E [ c + dX ] = c + d E [ X ] = c + d µ x d 2 Var [ X ] = d 2 σ 2 Var [ c + dX ] = x ⇒ σ Y = | d | σ x Remark ( Standardized Random Variable ) An important special case of the previous results is X − µ x = , Z σ x E ( Z ) = 0 for which : Var ( Z ) = 1 . P. Konstantinou (AUEB) Statistics for Business – II September 6, 2020 15 / 67

Random Variables and Probability/Density Distributions Specific Discrete Probability Distributions Bernoulli Distribution Consider only two outcomes: “ success ” or “ failure ”. Let p denote the probability of success, and 1 − p be the probability of failure. Define random variable X : x = 1 if success, x = 0 if failure. Then the Bernoulli probability function is P ( X = 0 ) = ( 1 − p ) and P ( X = 1 ) = p Moreover: � µ X = E ( X ) = x · P ( x ) = 0 · ( 1 − p ) + 1 · p = p all x � ( x − µ X ) 2 · P ( x ) σ 2 Var ( X ) = E [( X − µ X ) 2 ] = = X all x ( 0 − p ) 2 ( 1 − p ) + ( 1 − p ) 2 p = p ( 1 − p ) = P. Konstantinou (AUEB) Statistics for Business – II September 6, 2020 16 / 67