Expectation - $n$ balls taken out of a box

Question

From a box with n numbered with $1$ to $n$ balls we two times choose at random one ball with replacement. Let $X$ denote the biggest of the number which appeared on a chose balls. Find a distribution of $X$ and find $\mathbb{E}X$.

How to find that? I know that distribution is $$\frac{1}{n} P(Y \le k) {2 \choose 1} +\frac{1}{n^2}$$ but i have utterly no idea why.

How to find $\mathbb{E}X$?

Answer 1

Hints:

• What is the probability the first chosen ball is $k$ or less?
• What is the probability the second chosen ball is $k$ or less?
• What is the probability both chosen balls are $k$ or less? This is $\mathbb{P}(X \le k)$
• What is $\mathbb{P}(X=k)=\mathbb{P}(X \le k)-\mathbb{P}(X \le k-1)$?
• What is $\displaystyle \sum_{k=1}^n k \,\mathbb{P}(X=k)$?