Question

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

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

$\mathbb P(X=2)=\dbinom52\left(\dfrac14\right)^2\left(1-\dfrac34\right)^{5-2}=\dfrac{135}{512}\approx0.26$

$\mathbb P(2\le X\le4)=\displaystyle\sum_{x=2}^4\mathbb P(X=x)$
$=\dbinom52\left(\dfrac14\right)^2\left(1-\dfrac14\right)^{5-2}+\dbinom52\left(\dfrac14\right)^3\left(1-\dfrac14\right)^{5-3}+\dbinom52\left(\dfrac14\right)^4\left(1-\dfrac14\right)^{5-4}$
$=\dfrac{375}{1024}\approx0.37$

The expected value of $X\sim\mathrm{Bin}(n,p)$ is simply $np$, while the standard deviation is $\sqrt{np(1-p)}$. In this case, they are $\dfrac54=1.25$ and $\sqrt{\dfrac{15}{16}}\approx0.97$, respectively.