It's a question on the 1000 Exercise in probability.

Let $X_1, X_2, X_3$ be independent random variables talking values on positive integers,
and having mass function given by $P(X_i=x)=(1-p_i)p_i^{x-1}$, for x =1,2,3,..., i=1,2,3

$$P(X_1<X_2<X_3)=\frac{(1-p_1)(1-p_2) p_2 p_3^2}{(1-p_2 p_3)(1-p_1p_2 p_3)}$$

The solution on the book is as following

My question is from first line to second line in the solution,
index "k" disappears, I think it's by making k=j+1,
but I don't know where $(1-p_3)$ goes, Also, from 2nd to 3rd line of the solution,
I don't know how the term $(1-p_2p_3) $ in the denominator coming from.

Thanks for your time!!


1 Answer 1


Hint: $\sum_{i<j<k} = \sum_{i<j}\sum_{k=j+1}^\infty$.

What is $\sum_{k=j+1}^\infty p_3^{k-1}$? Key is that $1-p_3$ is in the denominator of that sum.

The same thing for the next step. What is $$\sum_{j=i+1}^\infty (p_2p_3)^{j-1}?$$

  • $\begingroup$ Ah~ I got it!! Thank you Thomas!! $\endgroup$
    – Lily
    Commented Oct 22, 2013 at 23:57

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