Answer:
Cov (X,Y) = 6
Step-by-step explanation:
hello,
Cov(X,Y) = E(XY) - E(X)E(Y)
we must first find E(XY), E(X), and E(Y).
since X is uniformly distributed on the interval (0,12), then E(X) = 6.
next we find the joint density f(x,y) using the formula
![f(x,y) = g(y|x)f_{X}(x)](https://tex.z-dn.net/?f=f%28x%2Cy%29%20%3D%20g%28y%7Cx%29f_%7BX%7D%28x%29)
this is because f is uniformly distributed on the the interval (0,12)
also since the conditional probability density of Y given X=x, is uniformly distributed on the interval [0,x], then
for 0≤y≤x≤12
thus
.
hence,
![E(X,Y)= \int\limits^{12}_{x=0} \int\limits^x_{y=o} xy\frac{1}{12x} \,dy dx](https://tex.z-dn.net/?f=E%28X%2CY%29%3D%20%5Cint%5Climits%5E%7B12%7D_%7Bx%3D0%7D%20%20%5Cint%5Climits%5Ex_%7By%3Do%7D%20xy%5Cfrac%7B1%7D%7B12x%7D%20%20%5C%2Cdy%20dx)
![E(X,Y)=\frac{!}{24} \int\limits^{12}_{x=0} x^2 \, dx = 24](https://tex.z-dn.net/?f=E%28X%2CY%29%3D%5Cfrac%7B%21%7D%7B24%7D%20%5Cint%5Climits%5E%7B12%7D_%7Bx%3D0%7D%20x%5E2%20%5C%2C%20dx%20%20%3D%2024)
also,
![E(Y) = \int\limits^{12}_{x=0} \int\limits^x_{y=0} y\frac{1}{12x} \, dydx](https://tex.z-dn.net/?f=E%28Y%29%20%3D%20%5Cint%5Climits%5E%7B12%7D_%7Bx%3D0%7D%20%20%5Cint%5Climits%5Ex_%7By%3D0%7D%20%20y%5Cfrac%7B1%7D%7B12x%7D%20%20%5C%2C%20dydx)
![E(Y)=\frac{1}{24}\int\limits^{12}_{x=0} {x} \, dx =3](https://tex.z-dn.net/?f=E%28Y%29%3D%5Cfrac%7B1%7D%7B24%7D%5Cint%5Climits%5E%7B12%7D_%7Bx%3D0%7D%20%7Bx%7D%20%5C%2C%20dx%20%20%3D3)
thus Cov(X,Y) = E(XY) - E(X)E(Y)
= 24 - (6)(3)
= 6