# 9.3 - Finding Distributions

9.3 - Finding Distributions

## Proposition

A moment-generating function uniquely determines the probability distribution of a random variable.

##### Proof

If the support $$S$$ is $$\{b_1, b_2, b_3, \ldots\}$$, then the moment-generating function:

$$M(t)=E(e^{tX})=\sum\limits_{x\in S} e^{tx} f(x)$$

is given by:

$$M(t)=e^{tb_1}f(b_1)+e^{tb_2}f(b_2)+e^{tb_3}f(b_3)+\cdots$$

Therefore, the coefficient of:

$$e^{tb_i}$$

is the probability:

$$f(b_i)=P(X=b_i)$$

This implies necessarily that if two random variables have the same moment-generating function, then they must have the same probability distribution.

## Example 9-3

If a random variable $$X$$ has the following moment-generating function:

$$M(t)=\left(\dfrac{3}{4}+\dfrac{1}{4}e^t\right)^{20}$$

for all $$t$$, then what is the p.m.f. of $$X$$?

#### Solution

We previously determined that the moment generating function of a binomial random variable is:

$$M(t)=[(1-p)+p e^t]^n$$

for $$-\infty<t<\infty$$. Comparing the given moment generating function with that of a binomial random variable, we see that $$X$$ must be a binomial random variable with $$n = 20$$ and $$p=\frac{1}{4}$$. Therefore, the p.m.f. of $$X$$ is:

$$f(x)=\dbinom{20}{x} \left(\dfrac{1}{4}\right)^x \left(\dfrac{3}{4}\right)^ {20-x}$$

for $$x=0, 1, \ldots, 20$$.

## Example 9-4

If a random variable $$X$$ has the following moment-generating function:

$$M(t)=\dfrac{1}{10}e^t+\dfrac{2}{10}e^{2t} + \dfrac{3}{10}e^{3t}+ \dfrac{4}{10}e^{4t}$$

for all $$t$$, then what is the p.m.f. of $$X$$?

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