Question
Use the central limit theorem to evaluate$$\frac{1}{(n-1)!} \int_0^n e^{-x} x^{n-1} d x$$Hint: consider $P\left[S_n<x\right]$ where $S_n$ is a sum of $n$ id unit exponentially distributed random variables.
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Use the result that for a nonnegative random variable $Y$, $$ E[Y]=\int_{0}^{x} P[Y>t\} d t $$ to show that for a nonnegative random variable $X$, $$ E\left[X^{n}\right]=\int_{0}^{x} n x^{n-1} P\{X>x\} d x $$ HINT: Start with $$ E\left[X^{n}\right]=\int_{0}^{x} P\left\{X^{n}>t\right\} d t $$ and make the change of variables $t=x n$.
Continuous Random Variables
Theoretical Exercises
Show that $$ \lim _{n \rightarrow \infty} e^{-n} \sum_{k=0}^{n} \frac{n^{k}}{k !}=\frac{1}{2} $$ by applying the central limit theorem to suitably chosen, independent, Poisson distributed random variables.
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