Topic 8 The Expected Value


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1 Topic 8 The Expected Value Functions of Random Variables 1 / 12
2 Outline Names for Eg(X ) Variance and Standard Deviation Independence Covariance and Correlation 2 / 12
3 Names for Eg(X ) If g(x) = x, then µ = EX is called variously the distributional mean, and the first moment. If g(x) = x k, then EX k is called the kth moment. These names were made in analogy to a similar concept in physics. The second moment in physics is associated to the moment of inertia. If g(x) = (x µ) k, then E(X µ) k is called the kth central moment. The most frequently used central moment is the second central moment σ 2 = E(X µ) 2 commonly called the (distributional) variance. We often write Var(X ) for the variance. 3 / 12
4 Variance and Standard Deviation σ 2 = Var(X ) = E(X µ) 2 = EX 2 2µEX + µ 2 = EX 2 2µ 2 + µ 2 = EX 2 µ 2. In analogy with the corresponding concept with quantitative data, we call σ the standard deviation. If we subtract the mean and divide by the standard deviation, the resulting random variable Z = X µ σ has mean 0 and variance 1. Z is called the standardized version of X. 4 / 12
5 Variance and Standard Deviation Exercise. Compute the variance and standard deviation for 1. a single Bernoulli trial, 2. a fair dice 3. the distance on a dart board. 4. Y = ax + b for constants a and b. Give the answer in terms of the variance or standard deviation of X. 5 / 12
6 Names for Eg(X ) The third moment of the standardized random variable [ (X ) ] µ 3 E σ is called the skewness. Random variables with positive skewness have a more pronounced tail to the density on the right. Random variables with negative skewness have a more pronounced tail to the density on the left. The fourth moment of the standard normal random variable is 3. The kurtosis compares the fourth moment of the standardized random variable to this value [ (X ) ] µ 4 E 3. σ Random variables with a negative kurtosis are called leptokurtic. Lepto means slender. Those with a positive kurtosis are called platykurtic. Platy means broad. 6 / 12
7 Names for Eg(X ). Expected values in the case of more than one random variable is based on the same concepts as for a single random variable. For example, for two discrete random variables X 1 and X 2 and for a real valued function g, we have Eg(X 1, X 2 ) = g(x 1, x 2 )f X1,X 2 (x 1, x 2 ) x 1 x 2 the expected value is based on the joint mass function f X1,X 2 (x 1, x 2 ). 7 / 12
8 Independence For the case in which the random variables are independent. Here, we have the factorization identity f X1,X 2 (x 1, x 2 ) = f X1 (x 1 )f X2 (x 2 ) for the joint mass function. Apply the formulate for Eg(X 1, X 2 ) to the product of functions g(x 1, x 2 ) = g 1 (x 1 )g 2 (x 2 ) to find that E[g 1 (X 1 )g 2 (X 2 )] = x 1 g 1 (x 1 )g 2 (x 2 )f X1,X 2 (x 1, x 2 ) x 2 = g 1 (x 1 )g 2 (x 2 )f X1 (x 1 )f X2 (x 2 ) x 1 x 2 = g 1 (x 1 )g 2 (x 2 )f X1 (x 1 )f X2 (x 2 ) x 1 x 2 ( ) ( ) = g 1 (x 1 )f X1 (x 1 ) g 2 (x 2 )f X2 (x 2 ) x 1 x 2 = E[g 1 (X 1 )] E[g 2 (X 2 )] 8 / 12
9 Covariance and Correlation Take X 1 and X 2 random variables with respective means µ 1 and µ 2. Then the variance of their sum Var(X 1 + X 2 ) = E[((X 1 + X 2 ) (µ 1 + µ 2 )) 2 ] = E[((X 1 + X 2 ) (µ 1 + µ 2 )) 2 ] = E[((X 1 µ 1 ) + (X 2 µ 2 )) 2 ] = E[(X 1 µ 1 ) 2 ] + 2E[(X 1 µ 1 )(X 2 µ 2 )] + E[(X 2 µ 2 ) 2 ] = Var(X 1 ) + 2Cov(X 1, X 2 ) + Var(X 2 ). where the covariance Cov(X 1, X 2 ) = E[(X 1 µ 1 )(X 2 µ 2 )] 9 / 12
10 Covariance and Correlation The definition of covariance is analogous to that for a sample covariance. The analogy continues for the correlation, ρ, defined for random variables X 1 and X 2, as ρ(x 1, X 2 ) = Cov(X 1, X 2 ) Var(X1 ) Var(X 2 ). We can also modify the argument used for sample covariance to see that 1 ρ(x 1, X 2 ) 1. Correlation ±1 occurs only when X and Y have a perfect linear association. 10 / 12
11 Covariance and1.2 Correlation If X 1 and X 2 are independent, then 1 Cov(X 1, X 2 ) = E[(X 1 µ 1 )(X 2 µ 2 )] = E[X 1 µ 1 ] E[X 2 µ 2 ] = 0 and Var(X 1 + X 2 ) = Var(X 1 ) + Var(X 2 ). We can generalize the Pythagorean identity to independent X 1,..., X n and constants c 1,..., c n ! X2! X1 +X 2 Var(c 1 X 1 + c n X n ) = c 2 1 Var(X 1) + + c 2 nvar(x n ). 0!0.2! X1 Figure: Pythagorean identity for the variance of independent random variables! / 12
12 Covariance and Correlation Exercise. 1. Let X and Z be independent random variables mean 0, variance 1. Define Y = ρ 0 X + 1 ρ 2 0 Z. Show that Y has mean 0, variance 1. Show that X and Y have correlation ρ Find the variance of a binomial random variable based on n trials with success parameter p. 12 / 12
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