What is MGF in statistics?

In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution. There are particularly simple results for the moment-generating functions of distributions defined by the weighted sums of random variables.

How do you calculate MGF?

The second central moment is the variance of X. Similar to mean and variance, other moments give useful information about random variables. The moment generating function (MGF) of a random variable X is a function MX(s) defined as MX(s)=E[esX].

What does MGF stand for in math?

Moment Generating Function (MGF) of a Random Vector Y: The MGF of an n × 1 random vector Y is given by. m Y ( t ) = E ( e t ′ Y )

How do you find the expected value of MGF?

For the expected value, what we’re looking for specifically is the expected value of the random variable X. In order to find it, we start by taking the first derivative of the MGF. Once we’ve found the first derivative, we find the expected value of X by setting t equal to 0.

How to find the value of the MGF?

Every consecutive derivative of the MGF gives you a different moment. Each moment is equal to the expected value of X raised to the power of the number of the moment. By taking the first derivative ( n = 1) of the MGF and setting t equal to 0, we find the expected value or mean of random variable X.

How to calculate the MGF of an exponential random variable?

The next example shows how the mgf of an exponential random variable is calculated. Example Let be a continuous random variable with support and probability density function where is a strictly positive number. The expected value can be computed as follows: Furthermore, the above expected value exists and is finite for any , provided .

Why is a probability distribution uniquely determined by its MGF?

A probability distribution is uniquely determined by its MGF. If two random variables have the same MGF, then they must have the same distribution. ( Proof.) For the people (like me) who are curious about the terminology “moments”: Why is a moment called moment?

How does the moment generating function ( MGF ) work?

The beauty of MGF is, once you have MGF (once the expected value exists), you can get any n-th moment. MGF encodes all the moments of a random variable into a single function from which they can be extracted again later. A probability distribution is uniquely determined by its MGF.