Finding the Probability distribution function given Moment Generating Function

[Mona1990]

Hi everyone,

So I am taking a statistics course and finding this concept kinda challenging. wondering if someone can help me with the following problem!

Suppose X is a discrete random variable with moment generating function
M(t) = 2/10 + 1/10e^t + 2/10e^(2t) + 3/10e^(3t) + 2/10e^(4t)
where t is a real number.

we want to find the probability function of X.

I know that M(t) = E(e^(tx)) = e^(tx)* f(x)
but not sure what to do from there.

Thanks for the help ^^

 

[statdad]

You know the definition of the mgf of a discrete random variable is

$$m_X(t) = \sum_{k=0}^n {P(X=k) e^{kt}}$$

(I'm assuming the values of X are 0, 1, 2, ..., k for some integer k).

Match the terms of your mgf with this general form.

 

[Mona1990]

hi,
so from matching i get P (X = 0 ) = 2/10, P (X=1) = 1/10...P(X=4) = 2/10
but i don't get how to find the probability function knowing these values.

 

[statdad]

hi,
so from matching i get P (X = 0 ) = 2/10, P (X=1) = 1/10...P(X=4) = 2/10
but i don't get how to find the probability function knowing these values.

Look again at the numbers you have in your first line.

 

[Mona1990]

sorry I don't get it , what line?

 

[statdad]

P (x = 0 ) = 2/10, p (x=1) = 1/10...p(x=4) = 2/10

 

[Mona1990]

is it:
f(x) = 2/10 if x is even , and x/10 if x is odd?
thanks for all your help!

 

[statdad]

You can give the distribution of a discrete r.v. as a table - one for the values, the other for the probabilities - you don't have to specify a "formula" for them.