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Var(Sn) probability generating function

I have X₁ X₂, ..... being i.i.d.r.v
S_n is the sum of them.

I have shown that the probability generating function of S_n is equal to
G_N(G_X(z))

How can I use this to calculate that the variance of S_N?

Many thanks!!

Reply 1

lubus

There is a standard result for the variance from a probability generating function, You need to find the first and second derivatives and then apply

Var(x) = g''(1)+g'(1)-[g'(1)]^2

Proof is in your lecture notes i supose, or a quick google yields http://www.cl.cam.ac.uk/teaching/0708/Probabilty/prob06.pdf

(edited 10 years ago)

Reply 2

DFranklin

Original post by lubus

There is a standard result for the variance from a probability generating function, You need to find the first and second derivatives and then apply

Var(x) = g''(1)+g(1)-[g'(1)]^2Typo here I think.

RHS should be

g''(1) + g'(1) - (g'(1))^2

Reply 3

TommyCricket

Yep I've got that formula thanks.

Not sure how to differentiate the product of the generating functions though

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Reply 4

DFranklin

Original post by TommyCricket

Yep I've got that formula thanks.

Not sure how to differentiate the product of the generating functions thoughIt's not a product, it's a composition of functions.

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Reply 5

TommyCricket