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I have that $$ X_2|X_1 = Bin(X_1,p)$$

and

$$ X_1 = Geo(\frac 1{B+1}) $$

I need to find the marginal distribution of $$X_2$$

I know that the joint distribution is given by $$P(X_1,X_2) = P(X_2|X_1)P(X_1)$$

I'm thinking to take the marginal I need to take the sum over X1 however it is getting messy very quickly.

$$ P(X_1,X_2) = {x_1 \choose x_2} p^{x_2}(1-p)^{x_1-x_2} \frac{1}{B+1}\left(\frac{B}{B+1}\right)^{x_1}$$

I know that to find the marginal for X2 I need to take the sum over X1. But I am stuck with the following.

$$ P(X_2) = \sum_{x_1=x_2}^\infty{x_1 \choose x_2} \left(\frac{p}{1-p}\right)^{x_2} \frac{1}{B+1}\left(\frac{B(1-p)}{B+1}\right)^{x_1}$$

Any hints or tips would be appreciated.

grand_chat
  • 38,951
hannah
  • 19

1 Answers1

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You are on the right track. To evaluate your sum, use the formula $$ \sum_{n=k}^{\infty} \binom{n}{k} x^n = \frac{x^k}{(1-x)^{k+1}} $$ which you can prove by induction.

grand_chat
  • 38,951