00:01
So let's find the mean of x.
00:09
So the mean of the random variable x is given by the first derivative of its moment generating function mgf.
00:19
So to find the mean we need to differentiate mx of t with respect to t and then evaluate it at t equals 0.
00:27
So we would get m prime of x of t equals 2e to the 2t plus 3e to the negative t.
00:48
And then when you evaluate that at t equals 0 we get m prime of x of 0 equals 2e to the 2 times 0 plus 3e to the negative 0 which equals 1 so x equals 1.
01:17
So secondly, let's find the variance of x.
01:26
So the variance of the of a random variable x is given by the second derivative of its mgf value at t equals 0.
01:34
So we need to differentiate m prime x of t with respect to t and then evaluate it at t equals 0.
01:45
So we would get m double prime x of t equals 4e to the 2t minus 3e to the negative t.
02:00
Now evaluating m triple prime x of t at t equals 0 we get m triple prime x of 0 equals 8e squared to the 0.
02:19
Times 0 sorry.
02:21
Plus 3e to the negative 0 which equals 11.
02:32
So x equals 11 for the skewness.
02:41
Sorry...