1. ## Standard Normal Distribution

~

How to show that and ~ gamma()? Also how to show that only the standard normal distribution satisfies these 2 properties? In other words, if you have and ~ gamma(), then ~ .

To show that :
I know that distribution of is , where . Also .
.
Hence .

To show that ~ gamma():
.

Now, this is where I am stuck. How do I continue to get gamma()? Also how to show if you have and ~ gamma(), then ~ .

2. ## Re: Standard Normal Distribution

I'm not a great theoretician, but I think there may be something wrong with the question because the gamma distribution usually has two arguments, alpha and beta. However, the distribution of one z^2 is chi square with 1 df, and the chi square is a simplified gamma with alpha = 2 and n = 2.beta with n = 1 in this case. It all looks hopeful. Perhaps if you work through the derivation of the chi square backwards you may get somewhere. kat

3. ## Re: Standard Normal Distribution

I did not learn chi square distribution so I think it can be found directly without resorting to going to chi square. I hope somebody who knows more about this will be able to help me.

4. ## Re: Standard Normal Distribution

Ok, differentiating with respect to gives me which is the probability density function of gamma. Hence, I believe I have solved it. Now, if someone could help me with the reverse direction, I would greatly appreciate it.

5. ## Re: Standard Normal Distribution

Originally Posted by geniusacamel
... Now, if someone could help me with the reverse direction, I would greatly appreciate it.

Well, in general, if a random variable has a gamma distribution then it could actually depend on three (positive) parameters – it is of the Pearson Type III system.

That said, most often, the two parameter system is used i.e. .

If you use the standardized form of the gamma distribution i.e. , then it tends to the standard normal distribution as the parameter tends to infinity:

,

for all real values of , where

is the standard normal cdf.

That said, similar results hold for the general gamma distribution as well as the special case of the family of chi-square distributions.

 Tweet

#### Posting Permissions

• You may not post new threads
• You may not post replies
• You may not post attachments
• You may not edit your posts