Answer:
a) 


b) 
for other case
c) ![E(Y_{(n)}) = \frac{n}{\theta^n} \frac{\theta^{n+1}}{n+1}= \theta [\frac{n}{n+1}]](https://tex.z-dn.net/?f=E%28Y_%7B%28n%29%7D%29%20%3D%20%5Cfrac%7Bn%7D%7B%5Ctheta%5En%7D%20%5Cfrac%7B%5Ctheta%5E%7Bn%2B1%7D%7D%7Bn%2B1%7D%3D%20%5Ctheta%20%5B%5Cfrac%7Bn%7D%7Bn%2B1%7D%5D)
![Var(Y_{(n)}) =\theta^2 [\frac{n}{(n+1)(n+2)}]](https://tex.z-dn.net/?f=%20Var%28Y_%7B%28n%29%7D%29%20%3D%5Ctheta%5E2%20%5B%5Cfrac%7Bn%7D%7B%28n%2B1%29%28n%2B2%29%7D%5D)
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:
![E(Y_{(n)}) = \frac{n}{\theta^n} \frac{\theta^{n+1}}{n+1}= \theta [\frac{n}{n+1}]](https://tex.z-dn.net/?f=E%28Y_%7B%28n%29%7D%29%20%3D%20%5Cfrac%7Bn%7D%7B%5Ctheta%5En%7D%20%5Cfrac%7B%5Ctheta%5E%7Bn%2B1%7D%7D%7Bn%2B1%7D%3D%20%5Ctheta%20%5B%5Cfrac%7Bn%7D%7Bn%2B1%7D%5D)
For the variance first we need to find the second moment like this:



And after evaluate we got:
![E(Y^2_{(n)}) = \frac{n}{\theta^n} \frac{\theta^{n+2}}{n+2}= \theta^2 [\frac{n}{n+2}]](https://tex.z-dn.net/?f=E%28Y%5E2_%7B%28n%29%7D%29%20%3D%20%5Cfrac%7Bn%7D%7B%5Ctheta%5En%7D%20%5Cfrac%7B%5Ctheta%5E%7Bn%2B2%7D%7D%7Bn%2B2%7D%3D%20%5Ctheta%5E2%20%5B%5Cfrac%7Bn%7D%7Bn%2B2%7D%5D)
And the variance is given by:
![Var(Y_{(n)}) = E(Y^2_{(n)}) - [E(Y_{(n)})]^2](https://tex.z-dn.net/?f=%20Var%28Y_%7B%28n%29%7D%29%20%3D%20E%28Y%5E2_%7B%28n%29%7D%29%20-%20%5BE%28Y_%7B%28n%29%7D%29%5D%5E2)
And if we replace we got:
![Var(Y_{(n)}) =\theta^2 [\frac{n}{n+2}] -\theta^2 [\frac{n}{n+1}]^2](https://tex.z-dn.net/?f=%20Var%28Y_%7B%28n%29%7D%29%20%3D%5Ctheta%5E2%20%5B%5Cfrac%7Bn%7D%7Bn%2B2%7D%5D%20-%5Ctheta%5E2%20%5B%5Cfrac%7Bn%7D%7Bn%2B1%7D%5D%5E2%20)
![Var(Y_{(n)}) =\theta^2 [\frac{n}{n+2} -(\frac{n}{n+1})^2]](https://tex.z-dn.net/?f=%20Var%28Y_%7B%28n%29%7D%29%20%3D%5Ctheta%5E2%20%5B%5Cfrac%7Bn%7D%7Bn%2B2%7D%20-%28%5Cfrac%7Bn%7D%7Bn%2B1%7D%29%5E2%5D)
And after do some algebra we got:
![Var(Y_{(n)}) =\theta^2 [\frac{n}{(n+1)(n+2)}]](https://tex.z-dn.net/?f=%20Var%28Y_%7B%28n%29%7D%29%20%3D%5Ctheta%5E2%20%5B%5Cfrac%7Bn%7D%7B%28n%2B1%29%28n%2B2%29%7D%5D)
Answer:
1). 16 π cm^2
2). 112 π cm^3
3). 351.68 cm^3
***If the explanation doesn’t make sense, let me know, and I’ll try to explain in a different way.
9514 1404 393
Answer:
y = 3x +6
Step-by-step explanation:
Put the numbers in the formula in their corresponding places.
y = mx + b . . . . slope-intercept form with slope m, intercept b
y = 3x +6 . . . . . slope-intercept form with slope 3, intercept 6
I am reading this correctly, the width = 1/3 of the length?
If this is the case, and we know that the area is 675, the equation is as follows.
Width x Length = 675 ft2
X = length
1/3 X x X = 675 ft2
To solve for X we have to multiply both sides by 3.
X square 2 = 2601 square feet.
Take the square of each side.
X = 225 sqare feet.
I hope this helps, let me know if you have any questions.