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}\frac1n&\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\frac xn`

Since
`\frac1n` is independent of
`x`, you have


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