00:01
Given that y is equal to n minus 1 s square divided by summation square where y belongs to chi square n minus 1 distribution, s square is the sample variance of identically distributed sample.
00:29
The sample follows normal distribution mu comma sigma square.
00:36
This is equal to chi square v random variable has m g t which is moment generating function at t which is capital m equal to t 1 minus 2 2 t power minus v by 2.
01:05
Now let us consider y belonging to chi square n minus 1.
01:13
Therefore, the moment generating function of y is m y t is equal to 1 minus 2 t.
01:28
This is 1 minus 2 t to the power minus n minus 1 divided by 2.
01:38
Therefore, mean can be calculated as t y is equal to the differentiation of moment generating function which is equal to d by dt with respect to t 1 minus 2 t to the power minus n minus 1 divided by 2.
02:02
This gives us n minus 1 the entire thing divided by 2 multiplied by 2 multiplied by 1 which is equal to n minus 1.
02:13
Therefore, t y is equal to n minus 1.
02:17
This is the part 1 of the question.
02:23
Again, t y square is equal to the double derivative of y t which is equal to d by dt of t y which in this case is equal to n square minus 1 multiplied by 1 minus 0 which is equal to n square minus 1...