Let X be a random variable, not necessarily positive.

(a) Using Markov's inequality, show that for x > 0 and t > 0,

assuming that E[e^tx] exists, where m is the mgf of X.

(b) For the case when X has a standard normal distribution, give the upper bound in Equation 9.5. Note that the bound holds for all t > 0.

(c) Find the value of t that minimizes your upper bound. If Z ∼ Norm (0, 1), show that for z > 0,

The upper bounds in Equations 9.5 and 9.6 are called Chern off bounds.