Moment Generating Function of normally distributed variable

In summary: The way I would do it (i.e. the second part) is to use the chi-square pdf and then use the mgf and do some algebra to get the final mgf function.
  • #1
johnaphun
14
0
Hi guys,

I need to find the moment generating function for X ~ N (0,1) and then also the MGF for X2 . I know how to do the first part but I'm unsure for X2.

do i use the identity that if Y = aX then

MY(t) = E(eY(t)) = E(e(t)aX)

or do i just square 2pi-1/2e x2/2 and then solve as normal for an MGF?
 
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  • #2
johnaphun said:
Hi guys,

I need to find the moment generating function for X ~ N (0,1) and then also the MGF for X2 . I know how to do the first part but I'm unsure for X2.

do i use the identity that if Y = aX then

MY(t) = E(eY(t)) = E(e(t)aX)

or do i just square 2pi-1/2e x2/2 and then solve as normal for an MGF?

Hey johnaphun and welcome to the forums.

For the chi-square distribution, are you given the pdf function or do you have to derive it from the normal function?

If you can use the definition of the pdf, then you do exactly the same thing as you did for the normal. If you need to derive the pdf for the chi-square distribution then you need to use a transformation method.

You don't need to use any identity, just the pdf of the appropriate distribution as well as the MGF definition.
 
  • #3
chiro said:
Hey johnaphun and welcome to the forums.

For the chi-square distribution, are you given the pdf function or do you have to derive it from the normal function?

If you can use the definition of the pdf, then you do exactly the same thing as you did for the normal. If you need to derive the pdf for the chi-square distribution then you need to use a transformation method.

You don't need to use any identity, just the pdf of the appropriate distribution as well as the MGF definition.

Hi thanks for the reply,

I've not been given any pdf function, I've only been told that Xi is standardised normally distributed and to thus find the MGF for Xi (which i can do) and (Xi)2.

I assume it's asking for me to find an MGF for a random variable Xi with chi distribution which is (1 − 2 t)−1/2, I'm just not sure how to go about proving that.
 
Last edited:
  • #4
johnaphun said:
Hi thanks for the reply,

I've not been given any pdf function, I've only been told that Xi is standardised normally distributed and to thus find the MGF for Xi (which i can do) and (Xi)2.

I assume it's asking for me to find an MGF for a random variable Xi with chi distribution which is (1 − 2 t)−1/2, I'm just not sure how to go about proving that.

That seems a little odd because at the very least you should be given the normal pdf function.

The way I would do it (i.e. the second part) is to use the chi-square pdf and then use the mgf and do some algebra to get the final mgf function.

The thing though is that if you have to derive the chi-square pdf from the normal pdf, then you may lose marks. It doesn't explicitly say you have to do this though, so you are probably in all likelihood, safe to just use the chi-square pdf.

Based on this information, do you know how to go about solving this problem?
 
  • #5


Hi there,

To find the moment generating function (MGF) for X ~ N(0,1), you can use the formula MY(t) = E(e^(tX)). In this case, t is the parameter in the MGF and X is your normally distributed variable. To find the MGF for X2, you can use the identity that if Y = aX, then MY(t) = E(e^(tY)) = E(e^(taX)). In this case, a = 2, so your MGF for X2 would be MY(t) = E(e^(2tX)). You can then use the formula for the MGF of a normally distributed variable to solve for the MGF of X2. This would involve using the standard deviation and mean of X2 in the formula, which are related to the standard deviation and mean of X. I hope this helps!
 

1. What is a moment generating function (MGF)?

A moment generating function (MGF) is a mathematical function that is used to characterize the probability distribution of a random variable. It is defined as the expected value of the exponential function raised to the power of the random variable.

2. How is the MGF of a normally distributed variable defined?

The MGF of a normally distributed variable is defined as M(t) = e^(μt + (σ^2)t^2/2), where μ is the mean and σ is the standard deviation of the normal distribution.

3. What is the importance of the MGF in statistics?

The MGF is important in statistics because it allows us to easily calculate moments of a probability distribution, such as the mean and variance. It also allows us to derive higher-order statistics, such as skewness and kurtosis, which can be used to further describe the shape of the distribution.

4. How is the MGF used to find the distribution of a sum of independent normally distributed variables?

The MGF of a sum of independent normally distributed variables can be found by taking the product of the individual MGFs. This allows us to easily find the MGF of more complex distributions, such as the sum of multiple random variables.

5. Can the MGF be used for non-normally distributed variables?

Yes, the MGF can be used for any probability distribution, not just the normal distribution. However, for some distributions, the MGF may not exist or may be difficult to calculate. In those cases, other methods may be used to characterize the distribution.

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