Two statements are given, one labelled Assertion (A) and the other labelled Reason (R).

Assertion (A): Suppose that X has poisson distribution.

If $P(X = 3) =\frac{3}{4} P(X = 2)$, then mean of X is $\frac{9}{4}$.

Reason (R): The recurrence formula for poisson distribution is $P(r + 1) =\frac{λ}{r+1}P(r),r=0,1,2,3...$ where $λ = np$.

Select the correct answer from the options given below.

Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).

The correct answer is Option (1) → Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).

We know that recurrence formula for poisson distribution is $P(r + 1) =\frac{λ}{r+1}P(r)$

∴ Reason is true.

Using recurrence formula, we have

$P(X = 3)=\frac{λ}{3}=P(X = 2)$.

Given $P(X = 3) =\frac{3}{4}P(X = 2)$

On comparing, we get

$\frac{λ}{3}=\frac{3}{4}⇒λ=\frac{9}{4}$

∴ Assertion is true.

Reason is the correct explanation of Assertion.