Respuesta :
Answer:
The value of Cov (X, Y) is 25/6.
Step-by-step explanation:
It is provided that:
[tex]X\sim U(0,10)\\\\Y|X\sim U(0,x)[/tex]
The probability density functions are as follows:
[tex]f_{X}(x)=\left \{ {{\frac{1}{10};\ 0<X<10} \atop {0;\ \text{otherwise}}} \right. \\\\f_{Y|X}(y|x)=\left \{ {{\frac{1}{x};\ 0<Y<x} \atop {0;\ \text{otherwise}}} \right.[/tex]
Then the value of f (x, y) will be:
[tex]f_{X,Y}(x,y)=\left \{ {{\frac{1}{10x};\ 0<X<10,\ 0<Y<x} \atop {0;\ \text{Otherwise}}} \right.[/tex]
Then f (y) is:
[tex]f_{Y}(y)=\int\limits^{10}_{y} {\frac{1}{10x}} \, dx[/tex]
[tex]=\frac{1}{10}\times [\log x]^{10}_{y}\\\\=\frac{1}{10}[\log 10-\log y][/tex]
Compute the value of E (X) as follows:
[tex]E(X)=\frac{b+a}{2}=\frac{10+0}{2}=5[/tex]
Compute the value of E (Y) as follows:
[tex]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}[/tex]
Compute the value of E (XY) as follows:
[tex]E(XY)=\int\limits^{10}_{0}\int\limits^{x}_{0} {xy\cdot \frac{1}{10x}} \, dx dy[/tex]
[tex]=\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}[/tex]
Compute the value of Cov (X, Y) as follows:
[tex]Cov (X, Y)=E(XY)-E(X)E(Y)[/tex]
[tex]=\frac{50}{3}-[5\times\frac{5}{2}]\\\\=\frac{50}{3}-\frac{25}{2}\\\\=\frac{100-75}{6}\\\\=\frac{25}{6}[/tex]
Thus, the value of Cov (X, Y) is 25/6.
