Choose a random subset of $\{2^1,2^2,⋯,2^{10}\}$ by selecting each of the 10 elements independently with probability $\frac{1}{2}$. Find the expected value of the smallest element in the subset (e.g. the subset can be $\{2^1,2^3,2^4,2^7\}$. The smallest element is $2^1$).
- I am not sure how to approach this question. I did however try to do $\{(2^1*1)+(2^2*2)+\cdots+(2^{10}*10)\}$ and did not get the right answer.
Any assistance would help.
Best Answer
The probability that $2$ is chosen is $\frac12$.
The probability that $2^2$ is the smallest element chosen is $\left(\frac12\right)^2$. We do not pick $2$ and then pick $2^2$.
In general, the probability of $2^k$ is the smallest element is $\left( \frac12 \right)^k$.
Hence it should be evaluated to be $$\sum_{k=1}^{10}2^k\cdot \left( \frac12\right)^k=10$$