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

Answer:

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

Step-by-step explanation:

It is provided that:

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

The probability density functions are as follows:

$f_{X}(x)=\left \{ {{\frac{1}{10};\ 0$

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

$f_{X,Y}(x,y)=\left \{ {{\frac{1}{10x};\ 0$

Then f (y) is:

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

         $=\frac{1}{10}\times [\log x]^{10}_{y}\\\\=\frac{1}{10}[\log 10-\log y]$

Compute the value of E (X) as follows:

$E(X)=\frac{b+a}{2}=\frac{10+0}{2}=5$

Compute the value of E (Y) as follows:

$E(Y|X)=\frac{b+a}{2}=\frac{x+0}{2}=\frac{x}{2}\\\\\text{Then,}\\\\E(E(Y|X))=E(\frac{x}{2})\\\\E(Y)=\frac{1}{2}\times E(X)\\\\E(Y)=\frac{5}{2}$

Compute the value of E (XY) as follows:

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

             $=\int\limits^{10}_{0}\int\limits^{x}_{0} {\frac{y}{10}} \, dx dy\\\\=\frac{1}{10}\times \int\limits^{10}_{0}{\frac{y^{2}}{2}}|^{x}_{0} \, dx \\\\=\frac{1}{10}\times \int\limits^{10}_{0}{\frac{x^{2}}{2}}\, dx\\\\=\frac{1}{10}\times [\frac{x^{3}}{6}]^{10}_{0}\\\\=\frac{100}{6}\\\\=\frac{50}{3}$

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

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

                 $=\frac{50}{3}-[5\times\frac{5}{2}]\\\\=\frac{50}{3}-\frac{25}{2}\\\\=\frac{100-75}{6}\\\\=\frac{25}{6}$

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