Explain why a gamma random variable with parameters $(t, \lambda)$ has an approximately normal distribution when $t$ is large.

Solution 1:

That's pretty much it except that you assumed $t$ is an integer and you used "$i$" rather then "$t$" at one point. One way of dealing with non-integer values of $t$ is to go back to the proof of the CLT that uses characteristic functions and make a minor modification in the argument to accomodate non-integers.

(BTW, one writes $Y\sim\mathrm{Exp}$, with that entire expression in MathJax and with \mathrm{} or the like, not $Y$ ~ $Exp$.)

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