Annual windstorm losses, X and Y, in two different regions are independent, and each is uniformly distributed on the interval [0
, 10]. Calculate the covariance of X and Y, given that X+ Y < 10.
1 answer:
Answer:

Step-by-step explanation:
Given
![Interval =[0,10]](https://tex.z-dn.net/?f=Interval%20%3D%5B0%2C10%5D)

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) =\frac{1}{100} [\frac{100X^2}{2} - \frac{20X^3}{3} + \frac{X^4}{4}]|\limits^{10}_0](https://tex.z-dn.net/?f=E%28XY%29%20%3D%5Cfrac%7B1%7D%7B100%7D%20%5B%5Cfrac%7B100X%5E2%7D%7B2%7D%20-%20%5Cfrac%7B20X%5E3%7D%7B3%7D%20%2B%20%5Cfrac%7BX%5E4%7D%7B4%7D%5D%7C%5Climits%5E%7B10%7D_0)
Expand
![E(XY) =\frac{1}{100} ([\frac{100*10^2}{2} - \frac{20*10^3}{3} + \frac{10^4}{4}] - [\frac{100*0^2}{2} - \frac{20*0^3}{3} + \frac{0^4}{4}])](https://tex.z-dn.net/?f=E%28XY%29%20%3D%5Cfrac%7B1%7D%7B100%7D%20%28%5B%5Cfrac%7B100%2A10%5E2%7D%7B2%7D%20-%20%5Cfrac%7B20%2A10%5E3%7D%7B3%7D%20%2B%20%5Cfrac%7B10%5E4%7D%7B4%7D%5D%20-%20%5B%5Cfrac%7B100%2A0%5E2%7D%7B2%7D%20-%20%5Cfrac%7B20%2A0%5E3%7D%7B3%7D%20%2B%20%5Cfrac%7B0%5E4%7D%7B4%7D%5D%29)
![E(XY) =\frac{1}{100} ([\frac{10000}{2} - \frac{20000}{3} + \frac{10000}{4}] - 0)](https://tex.z-dn.net/?f=E%28XY%29%20%3D%5Cfrac%7B1%7D%7B100%7D%20%28%5B%5Cfrac%7B10000%7D%7B2%7D%20-%20%5Cfrac%7B20000%7D%7B3%7D%20%2B%20%5Cfrac%7B10000%7D%7B4%7D%5D%20-%200%29)
![E(XY) =\frac{1}{100} ([5000 - \frac{20000}{3} + 2500])](https://tex.z-dn.net/?f=E%28XY%29%20%3D%5Cfrac%7B1%7D%7B100%7D%20%28%5B5000%20-%20%5Cfrac%7B20000%7D%7B3%7D%20%2B%202500%5D%29)

Take LCM


Calculate E(X)

Rewrite as:

Integrate Y

Expand
![E(X) =\frac{1}{50}\int\limits^{10}_0 ( [X*(10 - X)] - [X * 0])\ dX](https://tex.z-dn.net/?f=E%28X%29%20%3D%5Cfrac%7B1%7D%7B50%7D%5Cint%5Climits%5E%7B10%7D_0%20%28%20%5BX%2A%2810%20-%20X%29%5D%20-%20%5BX%20%2A%200%5D%29%5C%20dX)
![E(X) =\frac{1}{50}\int\limits^{10}_0 ( [X*(10 - X)]\ dX](https://tex.z-dn.net/?f=E%28X%29%20%3D%5Cfrac%7B1%7D%7B50%7D%5Cint%5Climits%5E%7B10%7D_0%20%28%20%5BX%2A%2810%20-%20X%29%5D%5C%20dX)

Integrate

Expand
![E(X) =\frac{1}{50}[(5*10^2 - \frac{1}{3}*10^3)-(5*0^2 - \frac{1}{3}*0^3)]](https://tex.z-dn.net/?f=E%28X%29%20%3D%5Cfrac%7B1%7D%7B50%7D%5B%285%2A10%5E2%20-%20%5Cfrac%7B1%7D%7B3%7D%2A10%5E3%29-%285%2A0%5E2%20-%20%5Cfrac%7B1%7D%7B3%7D%2A0%5E3%29%5D)
![E(X) =\frac{1}{50}[5*100 - \frac{1}{3}*10^3]](https://tex.z-dn.net/?f=E%28X%29%20%3D%5Cfrac%7B1%7D%7B50%7D%5B5%2A100%20-%20%5Cfrac%7B1%7D%7B3%7D%2A10%5E3%5D)
![E(X) =\frac{1}{50}[500 - \frac{1000}{3}]](https://tex.z-dn.net/?f=E%28X%29%20%3D%5Cfrac%7B1%7D%7B50%7D%5B500%20-%20%5Cfrac%7B1000%7D%7B3%7D%5D)

Take LCM


Calculate E(Y)

Rewrite as:

Integrate Y

Expand
![E(Y) =\frac{1}{50}\int\limits^{10}_0 ( [\frac{(10 - X)^2}{2}] - [\frac{(0)^2}{2}])\ dX](https://tex.z-dn.net/?f=E%28Y%29%20%3D%5Cfrac%7B1%7D%7B50%7D%5Cint%5Climits%5E%7B10%7D_0%20%28%20%5B%5Cfrac%7B%2810%20-%20X%29%5E2%7D%7B2%7D%5D%20-%20%5B%5Cfrac%7B%280%29%5E2%7D%7B2%7D%5D%29%5C%20dX)
![E(Y) =\frac{1}{50}\int\limits^{10}_0 ( [\frac{(10 - X)^2}{2}] )\ dX](https://tex.z-dn.net/?f=E%28Y%29%20%3D%5Cfrac%7B1%7D%7B50%7D%5Cint%5Climits%5E%7B10%7D_0%20%28%20%5B%5Cfrac%7B%2810%20-%20X%29%5E2%7D%7B2%7D%5D%20%29%5C%20dX)
![E(Y) =\frac{1}{50}\int\limits^{10}_0 [\frac{100 - 20X + X^2}{2}] \ dX](https://tex.z-dn.net/?f=E%28Y%29%20%3D%5Cfrac%7B1%7D%7B50%7D%5Cint%5Climits%5E%7B10%7D_0%20%5B%5Cfrac%7B100%20-%2020X%20%2B%20X%5E2%7D%7B2%7D%5D%20%5C%20dX)
Rewrite as:
![E(Y) =\frac{1}{100}\int\limits^{10}_0 [100 - 20X + X^2] \ dX](https://tex.z-dn.net/?f=E%28Y%29%20%3D%5Cfrac%7B1%7D%7B100%7D%5Cint%5Climits%5E%7B10%7D_0%20%5B100%20-%2020X%20%2B%20X%5E2%5D%20%5C%20dX)
Integrate
![E(Y) =\frac{1}{100}( [100X - 10X^2 + \frac{1}{3}X^3]|\limits^{10}_0)](https://tex.z-dn.net/?f=E%28Y%29%20%3D%5Cfrac%7B1%7D%7B100%7D%28%20%5B100X%20-%2010X%5E2%20%2B%20%5Cfrac%7B1%7D%7B3%7DX%5E3%5D%7C%5Climits%5E%7B10%7D_0%29)
Expand
![E(Y) =\frac{1}{100}( [100*10 - 10*10^2 + \frac{1}{3}*10^3] -[100*0 - 10*0^2 + \frac{1}{3}*0^3] )](https://tex.z-dn.net/?f=E%28Y%29%20%3D%5Cfrac%7B1%7D%7B100%7D%28%20%5B100%2A10%20-%2010%2A10%5E2%20%2B%20%5Cfrac%7B1%7D%7B3%7D%2A10%5E3%5D%20-%5B100%2A0%20-%2010%2A0%5E2%20%2B%20%5Cfrac%7B1%7D%7B3%7D%2A0%5E3%5D%20%29)
![E(Y) =\frac{1}{100}[100*10 - 10*10^2 + \frac{1}{3}*10^3]](https://tex.z-dn.net/?f=E%28Y%29%20%3D%5Cfrac%7B1%7D%7B100%7D%5B100%2A10%20-%2010%2A10%5E2%20%2B%20%5Cfrac%7B1%7D%7B3%7D%2A10%5E3%5D)


Recall that:



Take LCM


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