Question

The mean of binomial distribution is and the variance is 9/4.Evaluate PMF

Answer 1

If $X\sim Bin(n, p),$ then $E[X]=np, V[X]=np(1-p), \sigma_X=\sqrt{np(1-p)}.$

Given $E[X]=3, V[X]=\dfrac{9}{4}.$ Then

$$np=3$$

$$np(1-p)=\dfrac{9}{4}$$

$$1-p=\dfrac{3}{4}$$

$$p=\dfrac{1}{4}$$

$$n=12$$

$$f(x; 12,\dfrac{1}{4})= \begin{cases} \dbinom{12}{x}(\dfrac{1}{4})^x(1-\dfrac{1}{4})^{12-x},x=0, 1, ..., 12 \\ otherwise \end{cases}$$