91Ó°ÊÓ

A probability mass function is a cycle over the example space of a discrete arbitrary variable $X$ which gives the probability that $X$ is indistinguishable from a particular worth.

Let's define the random variable $Y$. It marks the largest number taken out of the urn. We find that $Y∈\{n,…,N\}$.

Let's take any $k∈\{1,…,N\}$.

From the information we observe that there are $\left(\begin{array}{l}N\\ n\end{array}\right)$of all possible combinations.

If the largest number taken out is $k$, we are capable to choose $n-1$out of $k-1$numbers freely.

Hence

$P(Y=k)=\frac{\left(\begin{array}{c}k-1\\ n-1\end{array}\right)}{\left(\begin{array}{c}N\\ n\end{array}\right)}$

Balls numbered $1$through $N$are in an urn. Suppose that $n,n≤N$, of them are randomly selected without replacement. Let $Y$denote the largest number select

We have that, $E\left(Y\right)=\underset{k=n}{\overset{N}{∑}} kâ‹\dots \frac{\left(\begin{array}{l}k−1\\ n−1\end{array}\right)}{\left(\begin{array}{l}N\\ n\end{array}\right)}=\frac{1}{\left(\begin{array}{l}N\\ n\end{array}\right)}\underset{k=n}{\overset{N}{∑}} k\left(\begin{array}{l}k−1\\ n−1\end{array}\right)$by using the definition of expectation.

Now, find that $k\left(\begin{array}{l}k−1\\ n−1\end{array}\right)=kâ‹\dots \frac{(k−1)!}{(n−1)!(k−n)!}=\frac{k!}{(n−1)!(k−n)!}=\frac{1}{n}\left(\begin{array}{l}k\\ n\end{array}\right)$

So we have the expression above is equal to $\frac{1}{\left(\begin{array}{l}N\\ n\end{array}\right)}\underset{k=n}{\overset{N}{∑}}\frac{1}{n}\left(\begin{array}{l}k\\ n\end{array}\right)=\frac{1}{n\left(\begin{array}{l}N\\ n\end{array}\right)}\underset{k=n}{\overset{N}{∑}}\left(\begin{array}{l}k\\ n\end{array}\right)$

Using Fermat's combinatoric identity, we have that

$\underset{k=n}{\overset{N}{∑}}\left(\begin{array}{l}k\\ n\end{array}\right)=\left(\begin{array}{l}N+1\\ n+1\end{array}\right)$

So finally, we have that

$E\left(Y\right)=\frac{1}{\left(\begin{array}{l}N\\ n\end{array}\right)}·\left(\begin{array}{c}N+1\\ n+1\end{array}\right)=\frac{N+1}{n(n+1)}$