Question

Let X and Y have the joint pdf f(x, y) = 3x, 0 < y < x < 1, zero elsewhere. Are X and Y independent? If not, find Var(Y|x).

Answer 1

$f(x,y)=3x,$ $0\lt y \lt x\lt1$

$0,\space elsewhere$

Given the $p. d. f$ above, we shall determine whether $X$ and $Y$ are independent using the property for testing independence given as,

$E(XY)=E(X)*E(Y)$

Now,

$E(XY)=\int^1_0\int^x_0 (xy)*f(x,y)dydx$

$=\int^1_0\int^x_0(xy)*3xdydx$

$=\int^1_0\int^x_03yx^2dydx$

$=$ $\int^1_03/2*(y^2x^2|^x_0)dx$

$=\int^1_0(3/2)*x^4dx$

$=(3/10*x^5)|^1_0=3/10$

$E(X) =\int^1_0\int^x_0xf(x,y)dydx$

$=\int^1_0\int^x_03x^2dydx$

$=\int^1_0(3x^2y)|^x_0dx$

$=\int^1_03x^3dx$

$=(3/4x^4)|^1_0$

$=3/4$

$E(Y)=\int^1_0\int^x_0y*f(x,y)dydx$

$=\int^1_0\int^x_03xydydx$

$=\int^1_03/2(xy^2)|^x_0dx$

$=\int^1_03/2(x^3)dx$

$=3/8(x^4)|^1_0$

$=3/8$

Let us check if the condition stated above is satisfied,

$3/10\not=3/4*3/8$

Therefore, random variables $X$ and $Y$ are not independent.

Since they are not independent, we move ahead to determine $V(Y|x)$.

To determine this conditional variance, we need to find the conditional distribution of $Y|x$ given as $f(Y|x)$ as follows,

$f(Y|x)=f(x,y)/g(x)$ where $g(x)$ is the marginal distribution of $X$

We already have $f(x,y)$, let's find $g(x)$

Now,

$g(x)=\int^x_0f(x,y)dy$

$=\int^x_03xdy$

$=(3xy)|^x_0$

$=3x^2$

Therefore, the marginal distribution of $X$ is given as,

$g(x)=3x^2,\space 0\lt x\lt 1$

$0,\space elsewhere$

Thus, $f(Y|x)=3x/3x^2=1/x,\space 0\lt y\lt x\lt1$

$0,\space elsewhere$

Conditional variance, $V(Y|x)=E(Y^2|x)-E(Y|x)$.

We determine the two conditional expectations as follows,

$E(Y|x)=\int^x_0y*f(Y|x)dy$

$=\int^x_0(y/x)dy$

$=(y^2/2x)|^x_0$

$=x^2/2x=x/2$

$E(Y^2|x)=\int^x_0y^2*f(Y|x)dy$

$=\int^x_0(y^2/x)dy$

$=(y^3/3x)|^x_0$

$=x^3/3x=x^2/3$

Now,

$V(Y|x)=x^2/3-(x/2)^2=x^2/12,\space 0\lt x\lt 1$

$0,\space elsewhere$