Question

a. Suppose X and Y are independent Poisson random variables, each with expected value 2. Define Z=X+Y. Find P(Z?3).

b. Consider a Poisson random variable X with parameter ?=5.3, and its probability mass function, pX(x). Where does pX(x) have its peak value?

Answer 1

(a)E(X)=2 and E(Y)=2, so mean of X is 2 and mean of Y is 2

this imply X~poisson(2) and Y~poisson(2)

if sum of two independent Poisson distributed random variables, with mean values ? and µ, also has Poisson distribution of mean ? + µ.

here Z=X+Y this imply Z~poisson(2+2). so Z will be poisson random variable with mean==4 and

P(X=x)=exp(-)^x/x!

P(Z3)=0.4334 ( using ms-excel =POISSON(3,4,1))

(b) required peak value=poisson(5)=0.173955

If ?<1 and P{X=0}>P{X=1}>P{X>2}? and so the mode is 0

• If ?>1 is not an integer, then the mode is ??? since
  P{X=???}<P{X=???}.

• If ?is an integer m, thenP{X=m}=P{X=m?1} and so either
  m or
  m?1 can be taken to be the mode.