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 ˆ¼ Norm(0,1), show that for z > 0,

The upper bounds in Equations 9.5 and 9.6 are called Chernoff bounds.

Transcribed Image Text:

E[e=] = e -t=m(t), P(X > x) < etr (9.5) P(Z > =)