Question

Random variable X is distributed Poisson, and P(X 2)-P(X 4). Find P(X 3)

Answer 1

answer is 0.2169

for Poisson distribution with parameter $\lambda$ the probability mass funciont

P(X=x)=exp(- $\lambda$ ) $\lambda$ ^{^x}/x!

here P(X=2)=P(X=4)

or, exp(- $\lambda$ ) $\lambda$ ^{^2}/2!=exp(- $\lambda$ ) $\lambda$ ^{^4}/4!

or, $\lambda$ ^{^2}/2= $\lambda$ ^{^4}/24

or, $\lambda$ ^{^2}=12 and $\lambda$ =sqrt(12)=3.464

now P(X=3)=exp(-3.464)(3.464)^{^3}/3!=0.2169

answer is 0.2169