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I want to show that:

$$\prod_{j}(1+a_{j})-\prod_{j}(1+b_{j})=\sum_{j}(\prod_{i < j}(1+a_{i})(a_{j}-b_{j})\prod_{k > j}(1+b_{k}))$$

The only hint I have is that I can replace $$ a_j - b_j = (1-a_j ) + ( 1-b_j) $$

I see there are $a_j$ and $-b_j$ terms after expansion of the RHS of the equation but I do not understand how the cross terms become zero.

user1916067
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    What is the statistical or probabilistic connection? Without some such this might be a better fit om math SE, ut as it is now, would probably be closed there, for lack of context – kjetil b halvorsen Jan 17 '24 at 11:00
  • The hint is slightly wrong, but you can use $a_j - b_j = (1+a_j) - (1+b_j)$. If you open the $j$th term in the RHS out with this, you should end up with a telescoping sum upon defining $u_j =\prod_{i \le j} (1+a_i) \prod_{k > j} (1+b_k).$ – stochasticboy321 Jan 17 '24 at 12:04
  • Another approach: because the LHS is a linear function of $a_1$ (with no constant term), you can identify its coefficient by differentiating. Do that to the RHS and see whether you can algebraically identify the result with what the LHS says. Your job might be made easier by working with $\alpha_j=1+a_j$ and $\beta_j=1+b_j$ as the variables. Staring at that for a few seconds might reveal clearly how the RHS telescopes into the LHS. – whuber Jan 17 '24 at 20:16

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