00:01
Um, yeah, so given here a sample size, um, n equals 3 from a uniform distribution on 0 theta, we can find the density of the sample mean.
00:09
So the sample mean, uh, for n equals 3 is simply the middle value of the three observations when they're ordered.
00:16
So the sample median, um, m is going to be the second order statistic, x sub 2.
00:20
And then for, um, n equals 3 from a uniform from distribution, we can use the density function of the ith order statistic, x sub i, that's going to be given by f xi of x is equal to n factorial divided by i minus 1 factorial factorial times n minus i factorial times x choose theta to the i minus 1 times 1 minus x over theta to the n minus i times 1 over theta.
01:07
So we get that for f of x sub 2 of x is going to be equal to, so it's going to be equal to, so 3 factorial, that's going to be 6 over 2 minus 1, which is is just 2.
01:22
Factorial is still going to be 2 times 3 minus 2.
01:26
Factorial is just, well, 6 over 1 times 1 factorial...