We are given a set X = {X_1, X_2, ..., X_n} where X_i = 2^i. A sample S is drawn by selecting each X_i independently wit...

Question

We are given a set X = {X_1, X_2, ..., X_n} where X_i = 2^i. A sample S is drawn by selecting each X_i independently with probability P = 1/2. The expected value of the smallest number in sample S is:

A. 1/n
B. 1/2
C. sqrt(n)
D. n/2

Accepted Answer

D. n/2 We have a set of n elements where the i-th element has value 2^i. Each element is independently included in sample S with probability 1/2. We need to find E[min(S)], the expected value of the smallest element selected. The key insight is to compute the probability that each element X_k = 2^k is the minimum of the sample, which requires that X_k is selected and all X_1, ..., X_{k-1} are not selected.

We are given a set X = {X_1, X_2, ..., X_n} where X_i = 2^i. A sample S is drawn by selecting each X_i independently with probability P = 1/2. The expected value of the smallest number in sample S is: A. 1/n B. 1/2 C. sqrt(n) D. n/2