Suppose that X is a continuous random variable with density function f. Show that E[|X − a|] is minimized when a is equal to the median of F.
Now break up the integral into the regions where x < a and where x > a, and differentiate.
Answer to relevant QuestionsThe Conditional Covariance Formula. The conditional covariance of X and Y, given Z, is defined by Cov(X, Y|Z) = E[(X − E[X|Z])(Y − E[Y|Z])|Z] (a) Show that Cov(X, Y|Z) = E[XY|Z] − E[X|Z]E[Y|Z] (b) Prove the conditional ...Let X1, . . . ,Xn be independent and identically distributed random variables. Find E[X1|X1 + · · · + Xn = x] Use the conditional variance formula to determine the variance of a geometric random variable X having parameter p. A.J. has 20 jobs that she must do in sequence, with the times required to do each of these jobs being independent random variables with mean 50 minutes and standard deviation 10 minutes. M.J. has 20 jobs that he must do in ...In Problem 7, suppose that it takes a random time, uniformly distributed over (0, .5), to replace a failed bulb. Approximate the probability that all bulbs have failed by time 550. Problem 7 A person has 100 light bulbs ...
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