Question

Determine whether the following are moment generating functions. If they are, give the corresponding probability distribution:

a) cosh(t)

b) sinh(t)

Answer 1

Answer:

Given,

a)

cosh(t)

Here Mx(t) is a moment generating function of a random variable

i.e.,

Mx(0) = 1

cosh0 = 1

cosht is a moment generating function

Now ,

For x = 1, Px(X) = 0.5

For x =-1, Px(X) = 0.5

So for the discrete distribution,

the moment generating function is Mx(t) = E(e^tx)

Mx(t) = 0.5e^t + 0.5e^-t

so Mx(t) = cosht

Hence our required pmf is Px(X) = 0.5 for x = +/- 1

b)

For sinh(t)

Here sinh0 = 0 i.e., not equal to 1

So that sinht can't be a moment generating function.