Joint Distribution?

Question

Engineering Stat Question

Let X be a Poisson random variable with lambda=2 and Y be a Poisson random variable with lambda=4. Both are independent. Determine the following probabilities

P(X<4,Y<4)

P(2<=X<4, Y >=3)

Answer 1

a. P(X<4,Y<4)=P(X<4) P(Y<4) ;because they are independent
=[P(x=0)+P(x=1)+P(X=2)+P(X=3)] [P(Y=0) + P(Y=1)+P(Y=2)+P(Y=3)]
=[(2^0 exp(-2))/0! + (2^1 exp(-2))/1! + (2^2exp(-2))/2! + (2^3 exp(-2))/3!] [(4^0 exp(-4)/0!) + (4^1 exp(-4)/1!) + (4^2 exp(-4))/2! + (4^3 exp(-2))/3!]
= use calculator to get the answer
(0!=1, 1!=1, 2!=2, 3!=6)

b. P(2<=X<4, Y>=3) = P(2<=X<4) P(Y>=3)
= [P(X=2)+P(X=3)] [1-P(Y<3)]
= [(2^2exp(-2))/2! + (2^3exp(-2))/3! ] [1-(4^0 exp(-4)/0!)-(4^1 exp(-4)/1!) - (4^2 exp(-4))/2! ]