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Q.
In a binomial distribution, mean is $3$ and standard deviation is $\frac{3}{2}$, then the probability function is
Probability - Part 2
Solution:
In a binomial distribution,
Mean $= np$ , Var = npq
and standard deviation $=\sqrt{ npq }$
$\therefore $ Mean $=3, S.D =\frac{3}{2}$
$\Rightarrow q =\frac{ npq }{ np }$ (multiply and divide by np)
$=\frac{9}{4 \times 3}=\frac{3}{4}$ $\left(\because \sqrt{ npq }=\frac{3}{2}\right)$
$\Rightarrow p =1-\frac{3}{4}=\frac{1}{4} (\because p =1- q )$
Now, $np =3$
$ \Rightarrow n \cdot \frac{1}{4}=3$
$ \Rightarrow n =12$
Therefore, binomial function is given as
$(q+p)^{n}=\left(\frac{3}{4}+\frac{1}{4}\right)^{12}$