Consider the following two-sided exponential PDF {

Chapter , Problem 28

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Consider the following two-sided exponential PDF { p)..e-AX , f X (x) = (1 _ p) .. e AX, if x 0, if x < 0, where ).. and p are scalars with )" > and p E [0, 1]. Find the mean and the variance of X in two ways: (a) By straightforward calculation of the associated expected values. (b) By using a divide-and-conquer strategy, and the mean and variance of the (onesided) exponential random variable. Solution. (a) and E[X] = f: xfx (x) dx = 1 x(1 - p) .. eAX dx + 100 xp)..e -AX dx - 00 = __ I _-_p + ).. ).. 2p - 1 - ).. E[X2 ] = f: x2 fx (x) dx = 1 x2(1 - p) .. eAX dx + 100 x2 p)..e-AX dx - 00 0 2(1 - p) 2p = )..2 + )..2 2 )..2' 2 (2P - 1 ) 2 var(X) = )..2 - ).. (b) Let A be the event {X O}, and note that P(A) = p. Conditioned on A, the random variable X has a (one-sided) exponential distribution with parameter )... Also, conditioned on AC the random variable -X has the same one-sided exponential distribution. Thus, and It follows that 1 E[X I A] = >:' E[X] = P(A)E[X I A] + P(AC)E[X l AC] -p 1 - p ).. ).. - -- 2p - l ).. E[X2 ] = P(A)E[X2 1 AI + P(AC)E[X2 1 AC] 2p 2(1 - p) = .;\2 + .;\2 2 - .;\2 ' 2 (2P - 1) 2 var(X) = .;\2 - .;\