Question

In Poisson probability distribution if P(r=2)= 3P(r=3), and P(r=4) is given by

A
e/24
B
24/e
С
1/(24e)
D
24e​

Answer 1

$\large\underline{\sf{Solution-}}$

We know,

Probability of any random variable 'r' having mean m using Poisson Distribution is given by

$\rm :\longmapsto\:P(r) = \dfrac{ {e}^{ - m} \: \: {m}^{r} }{r! }$

According to statement,

It is given that

P(2) = 3 P(3)

$\rm :\longmapsto\:\dfrac{ \cancel{ {e}^{ - m} }\: {m}^{2} }{2!} = 3 \dfrac{ \cancel{{e}^{ - m} }\: {m}^{3} }{3!}$

$\rm :\longmapsto\:\dfrac{1}{2 \times 1} = 3 \times \dfrac{m}{3 \times 2 \times 1}$

$\bf\implies \:m \: = \: 1$

Now,

$\rm :\longmapsto\:P(4)$

$\rm \: \: = \: \dfrac{ {e}^{ - m} \: {m}^{4} }{4!}$

$\rm \: \: = \: \dfrac{ {e}^{ - 1} {(1)}^{4} }{4!}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \red{\bigg \{ \because \:m \: = \: 1 \bigg \}}$

$\rm \: \: = \: \dfrac{ {e}^{ - 1} }{4 \times 3 \times 2 \times 1}$

$\rm \: \: = \: \dfrac{1}{24e}$

Hence,

$\bf :\longmapsto\:P(4) \: = \: \dfrac{1}{24e}$