00:01
I'm going to assume that x1 and x2 are independent.
00:07
Now for a, you want to find the variance of mu1 hat x1 over 2 plus x2 over 2.
00:18
So we will calculate the variance here.
00:35
So the variance of two independent random variables, or excuse me, the sum of two independent variables is the sum of the variance of these variables.
00:52
Now using the property of variance, we can factor out the constants.
01:06
Variance of x1 is given as 4, variance of x2 is given as 6, so that we have 5 over 2 or 2 .5.
01:19
Now for b we'll minimize the variance of this estimator by manipulating p so let's calculate the variance first i'm gonna use the same technique here and get 4 p squared plus 1 minus p squared 6 times 1 minus p squared excuse me is 1 minus p squared.
01:58
Now we're going to take the derivative of this variance with respect to p and set it equal to 0.
02:06
8p plus 12 times 1 minus p times negative 1.
02:17
So that we have 8p minus 12 plus 12p, set it equal to 0, and we get 12p equals 12, or p equals 3 over 5...