- Thread starter mh03
- Start date
- Tags expectation poisson proof variance

Hi! :welcome: We are glad that you posted here! This looks like a homework question though. Our homework help policy can be found here. We mainly just want to see what you have tried so far and that you have put some effort into the problem. I would also suggest checking out this thread for some guidelines on smart posting behavior that can help you get answers that are better much more quickly.

Specifically what have you tried so far?

Thank you for posting and letting me know about the protocol.

For question (a) I had this:

E(X)=

n

∑ x * p(x)

x=0

n

∑ x * λ^x * e^(-λ)/ x!

x=0

n

∑ x * λ^x * e^(-λ)/ (x-1)! * x

x=1

n

∑ λ^x * e^(-λ)/ (x-1)!

x=1

n

λ ∑ λ^(x-1) * e^(-λ)/ (x-1)! then let y= x-1

x=1

n

λ ∑ λ^y * e^(-λ)/ y!

y=0

= λ

For Question (b) i was having more trouble. I started with:

Var(X)= E(X^2) - E(X)^2

where we know E(X^2 - X) = E(X^2) - E(X)

Therefore: Var(X)= E[X(X-1)] + E(X) - E(X)^2 so we first need to solve E[X(X-1)]

E[X(X-1)]=

n

∑ x * (x-1) * λ^x * e^(-λ)/ x!

x=0

Any help with explanations and help as to where to go with these, or if i have made a mistake will be very much appreciated

Last edited:

n

∑ λ^x * e^(-λ)/ (x-1)!

x=1

**it was here that i had trouble and i got shown the next step, without explaination. I was wondering how the λ was factored across leaving λ^(x-1)?**

n

λ ∑ λ^(x-1) * e^(-λ)/ (x-1)! then let y= x-1

x=1

∑ λ^x * e^(-λ)/ (x-1)!

x=1

n

λ ∑ λ^(x-1) * e^(-λ)/ (x-1)! then let y= x-1

x=1

E[X(X-1)]=

n

∑ x * (x-1) * λ^x * e^(-λ)/ x!**this is where i got stuck**

x=0

Any help with explanations and help as to where to go with these, or if i have made a mistake will be very much appreciated

n

∑ x * (x-1) * λ^x * e^(-λ)/ x!

x=0

Any help with explanations and help as to where to go with these, or if i have made a mistake will be very much appreciated