Question

If $X$ is a binomial random variable with expected value $6$and variance$2.4$, find $P\{X=5\}$

Short answer

$P(X=5)=0.2$

Step-by-step solution

Step 1: Given information

Given in the question that,$X$is a binomial random variable with expected value$6$and variance $2.4,$

Step 2: Explanation

We have that $X~Binom(n,p)$.

We are given informations

$EX=6\Rightarrow np=6$

$Var\left(X\right)=2.4\Rightarrow np(1-p)=2.4$

Dividing the second equation with the first one, we end up with equation $1-p=\frac{2.4}{6}=0.4$, so we have that $p=0.6$. Hence, we also have that $n=10$.

Finally, we are required to determine following

$P(X=5)=\left(\begin{array}{c}10\\ 5\end{array}\right)0.6^{5}0.4^5=0.2$

Step 3: Final answer

$P(X=5)=0.2$