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Unformatted text preview: Expectations Enrique CamposN anez The George Washington University ApSc 116 1 / 41 Table of Contents The Expectation of a Random Variable Expectations and Joint Distributions Properties of the Expectations Expectation of Nonnegative r.v.s Variance Moments Covariance and Correlation Conditional Expectation Probability Theorems 2 / 41 Definition (Expectation) Let X be a discrete r.v. with p.f. f ( x ). We define the expectation of X as E ( X ) X x xf ( x ) , also called the mean of X . We say the expectation exists if X x  x  f ( x ) < . Example (A Single Die) Let X be the result of a single die toss. Then E ( X ) = X x xf ( x ) = 6 X x =1 x 1 6 = 1 6 [1 + 2 + 3 + 4 + 5 + 6] = 3 . 5 . 3 / 41 Example (Geometric Random Variable) Let 0 < p < 1, and let X be a random variable representing the number of attempts necessary to obtain a success in a Beroulli trial with success probability p . Then X is a geometric r.v., and its p.f. is f ( x ) = (1 p ) x 1  {z } x 1 failures p {z} success , x = 1 , 2 ,... We now compute its expected value or mean, i.e., E ( x ) = X x =1 x (1 p ) x 1 p = 1 p Note: this result requires the use of a moment generating function, which we will cover later in the semester. 4 / 41 Definition (Expectation of a Continuous r.v.) Let X be a continuous r.v. with p.d.f. f ( x ). We define its mean or expectation as E ( X ) Z  xf ( x ) dx . We say that E ( X ) exists in this case iff Z   x  f ( x ) dx < . Note that if there are numbers a , b such that < a < b < and Pr ( a X b ) = 1, then E ( X ) exists and a E ( X ) b . 5 / 41 Example (The Uniform ( a , b ) Distribution) Let X U ( a , b ). Then f ( x ) = 1 b a , and hence E ( X ) = Z b a x 1 b a dx = 1 b a Z b a xdx = 1 b a x 2 2 b x = a = b 2 a 2 2( b a ) = b + a 2 . 6 / 41 The Expectation of a function of a r.v. If X is a r.v. and Y = r ( X ), then Y is also a r.v. and we can calculate E ( Y ). It is not necessary to determine the probability distribution of Y to compute it. Continuous Case X is a continuous X r.v., and we say E ( Y ) exists iff R   r ( x )  f ( x ) dx < , and then we calculate E ( Y ) = Z  r ( x ) f ( x ) dx . Discrete Case Replace the integrals by sums, i.e., E ( Y ) = X x r ( x ) f ( x ) . 7 / 41 Example (Expected Project Delay Penalty) Suppose X is the random variable representing the duration in months of a project. Suppose X U (5 , 10). Suppose the company in charge of the project agreed to finish the project in 7 months, and a penalty of a 2 million dollars will be paid by this company for each additional month it takes to finish. What is the expected penalty? We can define the penalty as a new random variable, which is a function of X . Specifically, we define Y = ( 2( Y 7) if Y 7, otherwise....
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 Fall '08
 TomMazzuchi

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