Answer:

Explanation:
Given
![Interval =[0,10]](https://img.qammunity.org/2022/formulas/mathematics/college/t8zc496wukvfww1jj9mar1td4mxzat52ht.png)

Required

First, we calculate the joint distribution of X and Y
Plot

So, the joint pdf is:
--- i.e. the area of the shaded region
The shaded area is a triangle that has: height = 10; width = 10
So, we have:


is calculated as:

Calculate E(XY)


Make Y the subject

So, we have:

Rewrite as:

Integrate Y

Expand


Rewrite as:

Expand


Integrate
![E(XY) =(1)/(100) [(100X^2)/(2) - (20X^3)/(3) + (X^4)/(4)]|\limits^(10)_0](https://img.qammunity.org/2022/formulas/mathematics/college/6lnn9lbi2xul1nfcusjmbv43puex2laco5.png)
Expand
![E(XY) =(1)/(100) ([(100*10^2)/(2) - (20*10^3)/(3) + (10^4)/(4)] - [(100*0^2)/(2) - (20*0^3)/(3) + (0^4)/(4)])](https://img.qammunity.org/2022/formulas/mathematics/college/dw9yo7ejrcxjrru81pgn1mh3ce1xins7e5.png)
![E(XY) =(1)/(100) ([(10000)/(2) - (20000)/(3) + (10000)/(4)] - 0)](https://img.qammunity.org/2022/formulas/mathematics/college/fwr0wmaddm3ovywvybp4x2o7da548rea7i.png)
![E(XY) =(1)/(100) ([5000 - (20000)/(3) + 2500])](https://img.qammunity.org/2022/formulas/mathematics/college/q3u3ocitol8mfc2j93l2uzwu800qii5g2w.png)

Take LCM


Calculate E(X)

Rewrite as:

Integrate Y

Expand
![E(X) =(1)/(50)\int\limits^(10)_0 ( [X*(10 - X)] - [X * 0])\ dX](https://img.qammunity.org/2022/formulas/mathematics/college/z7kimvkfi4n4fto6ioane99ajbn1cuhw8f.png)
![E(X) =(1)/(50)\int\limits^(10)_0 ( [X*(10 - X)]\ dX](https://img.qammunity.org/2022/formulas/mathematics/college/atptalenfyihyihm7ajp774ibd5vkl1ra8.png)

Integrate

Expand
![E(X) =(1)/(50)[(5*10^2 - (1)/(3)*10^3)-(5*0^2 - (1)/(3)*0^3)]](https://img.qammunity.org/2022/formulas/mathematics/college/8qfzhalrbm4n3qwx6jzaexzx5ojwo3pm76.png)
![E(X) =(1)/(50)[5*100 - (1)/(3)*10^3]](https://img.qammunity.org/2022/formulas/mathematics/college/gokpju6tvte6fbfqa4sh0nz3briufenk90.png)
![E(X) =(1)/(50)[500 - (1000)/(3)]](https://img.qammunity.org/2022/formulas/mathematics/college/qox9psgbj9g4uhnoqq0ff3ob17ti18kgdi.png)

Take LCM


Calculate E(Y)

Rewrite as:

Integrate Y

Expand
![E(Y) =(1)/(50)\int\limits^(10)_0 ( [((10 - X)^2)/(2)] - [((0)^2)/(2)])\ dX](https://img.qammunity.org/2022/formulas/mathematics/college/l2ovuh3q73fvisdokqcbv12njhqlamjnay.png)
![E(Y) =(1)/(50)\int\limits^(10)_0 ( [((10 - X)^2)/(2)] )\ dX](https://img.qammunity.org/2022/formulas/mathematics/college/aiwkriisc4ajgz3u8rte4pvdz03l6f36o5.png)
![E(Y) =(1)/(50)\int\limits^(10)_0 [(100 - 20X + X^2)/(2)] \ dX](https://img.qammunity.org/2022/formulas/mathematics/college/sle9hsmk1jzll423kt04eiahqqeb9gyrla.png)
Rewrite as:
![E(Y) =(1)/(100)\int\limits^(10)_0 [100 - 20X + X^2] \ dX](https://img.qammunity.org/2022/formulas/mathematics/college/4t3zp141l61c3i7s1ufjxxfs6qq9tbriby.png)
Integrate
![E(Y) =(1)/(100)( [100X - 10X^2 + (1)/(3)X^3]|\limits^(10)_0)](https://img.qammunity.org/2022/formulas/mathematics/college/ab5htusdyx27440pe88l6b10belmbgtxjr.png)
Expand
![E(Y) =(1)/(100)( [100*10 - 10*10^2 + (1)/(3)*10^3] -[100*0 - 10*0^2 + (1)/(3)*0^3] )](https://img.qammunity.org/2022/formulas/mathematics/college/igfpt6pr940t83y7kxvnfkdb4emkqoa9lu.png)
![E(Y) =(1)/(100)[100*10 - 10*10^2 + (1)/(3)*10^3]](https://img.qammunity.org/2022/formulas/mathematics/college/53vh00k64advz8gpkvrj2wt8kkhqar5v75.png)


Recall that:



Take LCM

