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Serious? No clue. But I do know that my Mathematician and Statistician friends would be mad that you're confusing the two. Mods, change the title to "Stats geeks:..."
F you for knowing that. Now go slam yourself into a locker and give me your lunch money.
Since the poisson distribution is discrete, the proof isn't trivial. You just can't do the integral. You are gonna have to do the summations, and there is probably a summation trick or two in there to get it all looking nice and elegant. Set up the definition of expected value and variance and you will see this.
Also, for a math proof like this, you can't set them equal to each other. You solve each one of them individually, and show that the result is the same.
In this case, you need to find the find the mean in order to find the variance. And since the terms are sitting inside the summation, nothing will cancel.
rdawson i think your way is wrong but thats because i may have screwed up the original post.