(a) Let Z N(0, 1) and show that ~ Mz(t) =E[exp{tZ}] = exp {1}. t R....

 

Transcribed Image Text:

(a) Let Z N(0, 1) and show that ~ Mz(t) =E[exp{tZ}] = exp {1}. t R. Hint: This formula is obtained with a technique called "completion of a square", which is often used with Gaussian random variables. See pg. 106. (b) Let X ~N(1,02) and show that Mx(t) =E[exp{tX}] = exp ut P{t { ut + 0 2 1 }, t R. Hint: Use (a) and the fact that X can be expressed as X = +Z, where == Z~N(0, 1). (a) Let Z N(0, 1) and show that ~ Mz(t) =E[exp{tZ}] = exp {1}. t R. Hint: This formula is obtained with a technique called "completion of a square", which is often used with Gaussian random variables. See pg. 106. (b) Let X ~N(1,02) and show that Mx(t) =E[exp{tX}] = exp ut P{t { ut + 0 2 1 }, t R. Hint: Use (a) and the fact that X can be expressed as X = +Z, where == Z~N(0, 1).