Guest Chris123 Posted August 22, 2003 Share Posted August 22, 2003 Hey, I came across this question: Suppose X is a Poisson random quantity with parameter lambda>0 i.e suppose P[X=k}= e^-lambda * lambda^k /k! for k=0,1,2 etc a). derive a formula for E (X) in terms of lambda b). Show that E[X(X-1)] = lambda squared c). Derive a formula for Var(X) in terms of lambda For part a i'm assuming that the answer is just 'lambda' and also for part c since variance and E(X) is equal in the Poisson dist. and lambda = mean. Is this correct? However Im sttuck on part b. I deducted the following .... E(X(X-1)) = E(X^2 -X) = E(X)E(X) - E(X) but this equates to lambda^2 - lambda. where am i going wrong?? Thanks, Chris Link to comment Share on other sites More sharing options...
fafalone Posted August 22, 2003 Share Posted August 22, 2003 Les poissons, les poissons, how I love les poissons! Link to comment Share on other sites More sharing options...
Dave Posted August 22, 2003 Share Posted August 22, 2003 I spent a lot of time thinking about this until I realised it was quite easy. (btw, your answers to a and c are right) Now I don't know whether they want you to prove this from scratch, but my method is this: Var(X) = lambda = E(X^2) - [E(X)]^2 therefore: lambda + lambda^2 = E(X^2) From part (b), E(X^2 - X) = E(X^2) - E(X) = lambda^2 (by substitution). There may be another shorter method, I'm not entirely sure. Link to comment Share on other sites More sharing options...
Dave Posted August 24, 2003 Share Posted August 24, 2003 Originally posted by Chris123 a). derive a formula for E (X) in terms of lambda b). Show that E[X(X-1)] = lambda squared c). Derive a formula for Var(X) in terms of lambda okay, after thinking about this I realised that I was being completely stupid. The question itself is completely trivial when you just look at it. The key word here is 'derive' - hence you have to prove it yourself. here's a pdf with the answers in (too much time on my hands ). scistuff.pdf Link to comment Share on other sites More sharing options...
Recommended Posts
Create an account or sign in to comment
You need to be a member in order to leave a comment
Create an account
Sign up for a new account in our community. It's easy!
Register a new accountSign in
Already have an account? Sign in here.
Sign In Now