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+13 votes

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

  1. $\left(\frac{1}{n}\right)$
  2. $2$
  3. $\sqrt n$
  4. $n$
asked in Probability by Active (3.7k points)
edited by | 1.1k views

2 Answers

+31 votes
Best answer
The smallest element in sample $S$ would be $X_i$  for which $i$ is smallest.

The given probability is for selection of each item of $X$. Independent selection means each item is selected with probability $\frac{1}{2}$.

Probability for $X_1$ to be smallest in $S = \frac{1}{2} $.
Value of $X_1=2$.
Probability for $X_2$ to be smallest in $S$ = Probability of $X_1$ not being in S $\times$ Probability of $X_2$ being in $S$ $= \frac{1}{2} . \frac{1}{2} $.
Value of $X_2=2^2=4$.
Similarly, Probability for $X_i$ to be smallest in $S = (1/2)^i$.

Value of $X_i=2^i$ .

Now Required Expectation=  $\sum_{i=1}^{n}2^{^{i}} \times \left ( \frac{1}{2} \right )^{i} = \sum_{i=1}^n 1 = n $.

The answer is option D.
answered by Active (2k points)
selected by

What if S is ∅ ?

Well explained !
nice explanation thump up (y)

Superb .Thanks:)

–2 votes
Most probably 2 option b not sure.
answered by Active (3.3k points)
D is correct !

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