Recall the notions of expected value and variance. Given a random variable X, th
ID: 3297004 • Letter: R
Question
Recall the notions of expected value and variance. Given a random variable X, the true mean is E [X] and the true variance is Var [X]. If X has density f (x), then mu = E [X] = integral xf (x) dx and sigma^2 = Var [X] = integral (x - mu)^2 f (x)dx. Variance must be nonnegative. Expected value (or expectation) is linear. That is, for two random variables X and Y, E [X + Y] = E [X] + E [Y], and for a constant a, E [aX] = aE[X]. Variance is NOT linear. Var [aX] = a^2 Var [X]. Var [X + Y] = Var [X] + Var [Y] + 2 Cov[X, Y]. However, if X and Y are independent (or at least uncorrelated), then Cov [X, Y] = 0 and Var [X + Y] = Var [X] + Var[Y]. Suppose X and Y are independent random variables. Suppose E [X] = 5 and Var [X] = 4. Suppose E [Y] = 4 and Var [Y] = 1. Find: (a) E [ZX]. (b) E [2X + 4Y].(c) Var [2X + 4Y]. (d) Draw 100 realizations from a random normal with mean 5 and variance 4 (std dev 2) and store in x. Draw 100 realizations from a random normal with mean 4 and variance 1 and store in y. Report the mean and variance of 2*x + 4*y and compare to above. (e) Do expected value and variance live on the math side or the real world side of the bridge?Explanation / Answer
Suppose X and Y are independent random variables. And E[X] = 5, Var[X] = 4, E[Y] = 4, and Var[Y] = 1
a] E[3X] = 3 * E[X] = 3*5 = 15
b] E[2X + 4Y] = E[2x] + E[4Y] Since X and Y are independent
= 2*E[X] + 4*E[Y]
= 2*5 + 4*4
= 26
c] Var[2X + 4Y] = Var[2X] + Var[4Y] since X and Y independent Cov (X, Y) = 0
= 4*Var[X] + 16*Var[Y]
= 4* 4 + 16*1
= 32
d] here two random samples with E[x] = 5, Var[x] = 4, and E[y] = 4, Var[y] = 1
then Var[2x + 4y] = Var[2x] + Var[4y] since x and y random samples they are independent Cov (x, y) = 0
= 4*Var[x] + 16*Var[y]
= 4* 4 + 16*1
= 32
part c] and d] values are same but both have different meaning with respective to nature of random variables.
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