Question

A random variable $x$ follows binomial distribution with mean $3$and variance $\frac{3}{2}$, and $P(x\le 5)={\left(\frac{63}{2^x}\right)}^{}$Find the value of $x$

Answer 1

Let $n$,$p$ and $q$be the parameters of the binomial distribution.

Then,

Mean $=np=3$

Variance$=npq=\frac{3}{2}$

So,

$$\begin{gathered} \left(\frac{npq}{np}\right)=q \\ \Rightarrow \frac{\left({\displaystyle \frac{3}{2}}\right)}{3}=q \\ \Rightarrow q=\frac{1}{2} \end{gathered}$$

And,

$p=1–\frac{1}{2}=\frac{1}{2}$

Now, we have $p=\frac{1}{2}$and$np=3$

So,

$$\begin{gathered} np=3 \\ \Rightarrow n\times \frac{1}{2}=3 \\ \Rightarrow n=3\times 2=6 \end{gathered}$$

Thus, $n=6$

The binomial distribution is given by:

$P(X\le 5)=1-P(X=6)$

$$\begin{gathered} =1–{}^6{C}_6{\left(\frac{1}{2}\right)}^6{\left(\frac{1}{2}\right)}^{6-6} \\ =1–\frac{1}{64} \\ =\frac{63}{64} \\ =\frac{63}{2^6} \end{gathered}$$

Therefore, $x=6$