If a random variable X follows poisson distribution such that P(X=1) =2P(X=2) .find mean and the variance of the distribution .Also find P(X=3).

Subject: Applied Mathematics 2

Topic: Probability Distribution

Difficulty: High

Let the parameter of the Poisson distribution be m

∴ P(x) = $\frac{(e^{-m}×m^x)}{x!}$

Given that P(X=1) =2P(X=2)

∴ $\frac{(e^{-m}×m^1)}{1!}$ = 2 x $\frac{(e^{-m}×m^2)}{2!}$ ∴ m=1

∴ the mean & the variance = m = 1

Now P(X=3) = $\frac{(e^{-m}×m^x)}{x!}$ = $\frac{(e^{-1}×1^3)}{3!}$ = 0.0613.