Answer:
a)
b)
for other case
c)
Step-by-step explanation:
We have a sample of iid uniform on the interval and we want to find the cumulative distribution function.
Part a
For this case we can define the CDF for , like this:
Part b
For this case we know that:
And since are independent we have:
And then we can find the density function calculating the derivate from the last expression and we got:
for other case
Part c
For this case we can find the mean with the following integral:
And after evaluate we got:
For the variance first we need to find the second moment like this:
And after evaluate we got:
And the variance is given by:
And if we replace we got:
And after do some algebra we got: