Question

8. For each of the following, identify the distribution based on the MGF. Be sure to specify name of distribution and value(s) of parameter(s).

(a) MX(t) = (0.3 + 0.7 e t ) 9 , t ∈ (−∞,∞).

(b) MX(t) = 0.8 e t + 0.2, t ∈ (−∞,∞).

(c) MX(t) = e 9(e t−1), t ∈ (−∞,∞).

(d) MX(t) = 0.75e t 1−0.25e t , t < − ln 0.25.

(e) MX(t) = 0.4e t 1−0.6e t 20, t < − ln 0.6.

Answer 1

Answer:

Given that,

For each of the following, identify the distribution based on the MGF. Be sure to specify name of distribution and value(s) of parameter(s).

(a).

, t ∈ (−∞,∞)

Let p=0.7, q=1-0.7=0.3, n=9

The Standard is of form,

Which is M.G.F of Binomial of distribution with parameters n & p.

i.e, X Binomail(n,p) or X Binomial (9,0.7).

(b).

, t ∈ (−∞,∞).

Here, q=0.2, p=0.8

then,

Which is M.G.F of Binomial of distribution with parameters n & p.

i.e, X Binomail(n,p) or X Binomial (1,0.8)

(c).

, t ∈ (−∞,∞).

Which is the form of

Comparing both equation =9.

Which is M.G.F of Poisson of distribution with parameters .

i.e, X Poisson() or X Poisson (9).

(d).

, t < − ln 0.25.

It is the form of

Where p=0.75 and q=1-p=1-0.75=0.25

Which is the M.G.F of geometric (p) then, X Geo (0.75).

(e).

, t < − ln 0.6.

Let r=2, p=0.6

Then, the M.G.F of negative Binomial is

then X Negative Binomial(2,0.6).