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(Solved): Need part a, b, c and d Consider the random variable \( X \) with two-sided exponential distributi ...
![Consider the random variable \( X \) with two-sided exponential distribution given by
\[
f_{X}(x)=\frac{1}{2} e^{-|x|}
\]
wit](https://media.cheggcdn.com/media/803/803fb374-4afa-43aa-b5a2-b52219c131c8/phpFtW6uu)
Need part a, b, c and d
Consider the random variable \( X \) with two-sided exponential distribution given by \[ f_{X}(x)=\frac{1}{2} e^{-|x|} \] with corresponding moment generating function, \( M_{X}(s)=\frac{1}{1-s^{2}} \). (a) Using \( M_{X}(s) \) or otherwise, find the mean and variance of \( X \). (b) Use Chebychev inequality to estimate the tail probability \( P(X>\delta) \), for \( \delta>0 \). (c) Use the central limit theorem to estimate the tail probability \( P(X>\delta) \), for \( \delta>0 \). (d) Compare the Cherneff beund with Find the exact tail probability \( P(X>\delta) \), for \( \delta>0 \)