Question

Obtain the standard deviation of a random variable X whose moment generating function is 3/3-t.​

Answer 1

Answer:

In the discrete case mX is equal to P

x

e

txp(x) and in the continuous case ∞

−∞ e

txf(x)dx.

Let us compute the moment generating function for some of the distributions we have been

working with.

Example 13.1 (Bernoulli).

mX (t) = e

0·t

(1 − p) + e

1·t

p = e

t

p + 1 − p.

Example 13.2 (Binomial). Using independence,

Ee

t

PXi = E

Ye

tXi =

YEe

tXi = (pet + (1 − p))n

,

where the Xi are independent Bernoulli random variables. Equivalently

Xn

k=0

e

tk