Question

X and Y are modeled inflation rates, in terms of %, for two different countries at the end of a five-year period. X is uniformly distributed on the interval (0, 10). Y, given X = x is uniformly distributed on the interval (0, x). Calculate Cov(X, Y).

Answer 1

The value of Cov (X, Y) is 25/6.

Explanation:

It is provided that:

`X\sim U(0,10)\\\\Y|X\sim U(0,x)`

The probability density functions are as follows:

`f_(X)(x)=\left \{ {{(1)/(10);\ 0<X<10} \atop {0;\ \text{otherwise}}} \right. \\\\f_(Y|X)(y|x)=\left \{ {{(1)/(x);\ 0<Y<x} \atop {0;\ \text{otherwise}}} \right.`

Then the value of f (x, y) will be:

`f_(X,Y)(x,y)=\left \{ {{(1)/(10x);\ 0<X<10,\ 0<Y<x} \atop {0;\ \text{Otherwise}}} \right.`

Then f (y) is:

`f_(Y)(y)=\int\limits^(10)_(y) {(1)/(10x)} \, dx`

`=(1)/(10)* [\log x]^(10)_(y)\\\\=(1)/(10)[\log 10-\log y]`

Compute the value of E (X) as follows:

`E(X)=(b+a)/(2)=(10+0)/(2)=5`

Compute the value of E (Y) as follows:

`E(Y|X)=(b+a)/(2)=(x+0)/(2)=(x)/(2)\\\\\text{Then,}\\\\E(E(Y|X))=E((x)/(2))\\\\E(Y)=(1)/(2)* E(X)\\\\E(Y)=(5)/(2)`

Compute the value of E (XY) as follows:

`E(XY)=\int\limits^(10)_(0)\int\limits^(x)_(0) {xy\cdot (1)/(10x)} \, dx dy`

`=\int\limits^(10)_(0)\int\limits^(x)_(0) {(y)/(10)} \, dx dy\\\\=(1)/(10)* \int\limits^(10)_(0){(y^(2))/(2)}|^(x)_(0) \, dx \\\\=(1)/(10)* \int\limits^(10)_(0){(x^(2))/(2)}\, dx\\\\=(1)/(10)* [(x^(3))/(6)]^(10)_(0)\\\\=(100)/(6)\\\\=(50)/(3)`

Compute the value of Cov (X, Y) as follows:

`Cov (X, Y)=E(XY)-E(X)E(Y)`

`=(50)/(3)-[5*(5)/(2)]\\\\=(50)/(3)-(25)/(2)\\\\=(100-75)/(6)\\\\=(25)/(6)`

Thus, the value of Cov (X, Y) is 25/6.