Question

If $x$ is a random poission variable such that $B=P(X=2)=P(X=3)$, then $P(X=4)$ equals

• A. $2B$
• B. $\frac{3B}{4}$
• C. $\frac{3B^2}{4}$
• D. None of these

Answer 1

Answer: (B)

$\frac{3B}{4}$

Say m be the mean of the P.D. then
$p(X=r)=\frac{m^r{e}^{-m}}{r!},r=0,1,2,\cdots$

Now $\Rightarrow B=\frac{m^2{e}^{-m}}{2!}=\frac{m^3{e}^{-m}}{3!}\Rightarrow m=3$

now $B=P(X=2)=B\frac{3^2{e}^{-3}}{2!}=\frac{9}{2e^3}$

$\therefore P(X=4)=\frac{3^4{e}^{-3}}{4!}=\frac{27}{8e^3}=\frac{3B}{4}$