(Solved): Consider the random variable \( X \) with a triangular distribution given by \[ f_{X}(x)=\left\{\be ...

Consider the random variable \( X \) with a triangular distribution given by \[ f_{X}(x)=\left\{\begin{array}{rr} x+1, & -1 \leq x \leq 0 \\ 1-x, & 0 \leq x \leq 1 \end{array}\right. \] with corresponding moment generating function, \( M_{X}(k)=\frac{e^{t}+e^{-t}-2}{t^{2}} \). (a) Using \( M_{X}(t) \) or otherwise, find the mean and variance of \( X \). (b) Use Chebyshev inequality to estimate the tail probability \( P(X>\kappa) \), for \( \kappa>0 \). (C) Use-the-Chernoff bound te-stimatethetail-probability \( P(X>x) \), for \( x>0 \). (d) Use the central limit theorem to estimate tail probability \( P(X>\kappa) \), for \( \kappa>0 \). (e) Calculate the exact tail probability \( P(X>\kappa) \), for \( \kappa>0 \).