Question

The n candidates for a job have been ranked 1, 2, 3, …, n. let x = the rank of a randomly selected candidate, so that x has the following:
p(x = 1/n x=1,2,3,...,n
0 otherwise
a compute e(x.
there is a part b i will post serperately

Answer 1

$p(x)=\begin{cases}\dfrac1n&\text{for }x\in\{1,2,\ldots,n\}\\\\0&\text{otherwise}\end{cases}$

describes a discrete uniform distribution. The expectation is given by

$\mathbb E(X)=\displaystyle\sum_xxp(x)=\sum_{x=1}^nxp(x)=\sum_{x=1}^n\dfrac xn$

Since $\dfrac1n$ is independent of $x$, you have

$\displaystyle\sum_{x=1}^n\frac xn=\frac1n\sum_{x=1}^nx=\frac{\dfrac{n(n+1)}2}n=\dfrac{n+1}2$