# Finding the Probability distribution function given Moment Generating Function

• Mona1990
In summary, the conversation is about finding the probability function of a discrete random variable with a given moment generating function. The first speaker is struggling with understanding the concept and asks for help. The second speaker explains the definition of the moment generating function and how to match it with the general form. The third speaker suggests looking again at the numbers in the first line to find the probability function. The first speaker then asks for clarification and suggests a possible formula for the distribution of the random variable. The second speaker clarifies that a table can also be used to represent the distribution.
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 ^^

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.

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.

Mona1990 said:
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.

sorry I don't get it , what line?

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

is it:
f(x) = 2/10 if x is even , and x/10 if x is odd?

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.

## 1. What is a probability distribution function?

A probability distribution function (PDF) is a mathematical function that describes the probability of a random variable taking on a certain value or falling within a certain range of values. It is often represented graphically as a curve and is used to analyze the likelihood of different outcomes in a given situation.

## 2. What is a moment generating function?

A moment generating function (MGF) is a mathematical function that uniquely determines the probability distribution of a random variable. It is used to find the moments of a probability distribution, which are used to calculate important statistical properties such as the mean and variance.

## 3. How is the probability distribution function related to the moment generating function?

The moment generating function is related to the probability distribution function through a mathematical transformation. The MGF is the Fourier transform of the PDF, meaning that it provides a different representation of the same information. This allows us to find the PDF given the MGF.

## 4. Can the moment generating function always be used to find the probability distribution function?

Yes, the moment generating function can always be used to find the probability distribution function. However, this process can be complex and may require advanced mathematical techniques. In some cases, it may be more efficient to use other methods to find the PDF directly.

## 5. What is the importance of finding the probability distribution function given the moment generating function?

Finding the probability distribution function given the moment generating function allows us to better understand the underlying distribution of a random variable. This information is crucial in statistical analysis and decision-making, as it helps us to make predictions and draw conclusions about a particular situation or phenomenon.

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