Finding expected value from moment generating function

In summary, a moment generating function is a mathematical function that describes the distribution of a random variable. It is calculated by taking the expected value of e^tx, with t as a real number. The expected value can be found by taking the derivative of the moment generating function at t=0, also known as the first moment of the distribution. This function is useful for finding expected values without using complex integration or summation techniques, and can also help calculate higher moments of the distribution. It can be used for any type of distribution, as long as the distribution is defined by its probability density function. The moment generating function is closely related to other statistical measures, such as the mean and variance, and can also be used to find properties of
  • #1
jaejoon89
195
0
Find E(X) given the moment generating function

M_X (t) = 1 / (1-t^2)

for |t| < 1.

(The pdf is f(x) = 0.5*exp(-|x|), for all x, so graphically you can see that E(X) should be 0.)

----

I know that E(X) = M ' _X (t) = 0

BUT M ' _X (t) = 2x / (1-x^2)^2 which is indeterminate at 0 so maybe you need L'Hopital's rule or something but I can't get it to work. How do you do this?
 
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  • #2
What makes you say it's indeterminate at 0?
 

FAQ: Finding expected value from moment generating function

1. What is a moment generating function?

A moment generating function is a mathematical function that is used to describe the distribution of a random variable. It is defined as the expected value of e^tx, where t is any real number.

2. How is the expected value calculated from a moment generating function?

The expected value is calculated by taking the derivative of the moment generating function at t=0. This is also known as the first moment of the distribution.

3. Why is the moment generating function useful for finding expected value?

The moment generating function allows us to easily calculate the expected value of a random variable without having to use complicated integration or summation techniques. It also provides a way to find higher moments of the distribution.

4. Can the moment generating function be used for any type of distribution?

Yes, the moment generating function can be used for any type of distribution, as long as the distribution is defined by its probability density function. However, in some cases, the moment generating function may not exist or may be difficult to calculate.

5. How does the moment generating function relate to other statistical measures?

The moment generating function is closely related to other statistical measures such as the mean, variance, and higher moments of a distribution. In fact, the derivatives of the moment generating function at t=0 correspond to these measures. Additionally, the moment generating function can be used to find other properties of a distribution, such as the cumulative distribution function.

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