Question

For X a binomial random variable with n=5 and p=1/4, answer the following

A) what's P(X=2)

B) what's P(2
`\leq` X
`\leq` )

C) what's the expected value of X

D) what's the standard deviation of X

Answer 1

Recall that for
`X\sim\mathrm{Bin}(n,p)`, i.e. a random variable
`X` following a binomial distribution over
`n` trials and with probability parameter
`p`,


`\mathbb P(X=x)=f_X(x)=\begin{cases}\dbinom nx p^x(1-p)^(n-x)&\text{for }x\in\{0,1,\ldots,n\}\\\\0&\text{otherwise}\end{cases}`

So you have


`\mathbb P(X=2)=\dbinom52\left(\frac14\right)^2\left(1-\frac34\right)^(5-2)=(135)/(512)\approx0.26`


`\mathbb P(2\le X\le4)=\displaystyle\sum_(x=2)^4\mathbb P(X=x)`

`=\dbinom52\left(\frac14\right)^2\left(1-\frac14\right)^(5-2)+\dbinom52\left(\frac14\right)^3\left(1-\frac14\right)^(5-3)+\dbinom52\left(\frac14\right)^4\left(1-\frac14\right)^(5-4)`

`=(375)/(1024)\approx0.37`

The expected value of
`X\sim\mathrm{Bin}(n,p)` is simply
`np`, while the standard deviation is
`√(np(1-p))`. In this case, they are
`\frac54=1.25` and
`\sqrt{(15)/(16)}\approx0.97`, respectively.