00:01
An example 4 .13, we found that the moment generating function for a gamma distribution is equal to 1 minus beta t raised to the negative alpha power.
00:20
Now, this is the same as 1 over 1 minus beta t raised to the alpha like this.
00:26
You can see it written either way, but since we're going to need to differentiate this, i'm going to write it like we have it here on the left.
00:32
Now, we want to find the expected value and the variance by using this moment generating function.
00:38
Now the expected value is equal to the derivative of the moment generating function evaluated at zero.
00:49
Now our derivative here, using the chain rule is equal to negative alpha times one minus beta t to the negative alpha minus 1 times negative beta.
01:07
And so this is equal to alpha beta times 1 minus beta t to the negative alpha minus 1.
01:18
Evaluating this at zero gives us alpha beta times one minus zero to the negative alpha minus one one to the negative alpha minus one is just one and so this just gives us alpha beta and so the expected value is alpha beta now for the variance we're going to need our second derivative and so taking the second derivative here just means take the derivative again and so this is alpha beta times negative alpha minus 1 times 1 minus beta t to the negative alpha minus 2 times negative beta remember you're differentiating with respect to t and so we just took the derivative again with respect to t now the expected value of x squared is equal to the second derivative evaluated at 0 which is alpha beta negative alpha minus 1 times 1 minus 0 to the negative alpha minus 2 times negative beta.
02:39
Now negative alpha minus 1 times negative beta will make that positive...