I just have a few questions about joint density and marginal density questions.
Q1: Joint Distribution $f_1=2x+4y$ on triangle with vertices $(0,0), (0,1),(1,0)$. Sketch the region and compute marginal density of $X$. Use this to compute the expected value of $X$. Find the conditional density of Y given X. If X is .5, what is the expected value of $Y$. Lastly, the total is $Z=X+Y$. Find the density of Z.
The region is clearly just the triangle with those vertices. The marginal density is the integral of $f_1$ from $(0)$ to $(1-x)$ $dy$. The expected value is simply integral from 0 to 1 of $xf_{1x}(X)=\frac{1}{3}$.
The conditional density is simply the joint density divided by the marginal density of $X$. This I found to be $$\frac{2x+4y}{2-2x}=\frac{x+2y}{1-x}$$
Therefore $E(Y|X=.5)=\int^{.5}_0 yf_{y|x}=\frac{7}{24}$
Lastly, the cumulative distribution function of $X+Y$ I determined as $$\int_0^z\int^{z-x}_0f(x,y)dydx=\int_0^z\int^{z-x}_0(2x+4y)dydx=\int_0^z((2xy+2y^2))|^{z-x}_0=\int_0^z2x(z-x)+2(z-x)^2$$ $$=\int_0^z2xz-2x^2+z^2-2zx+x^2=\int_0^z-x^2+z^2=(\frac{-x}{3}+xz^2)|^z_0=\frac{-z^3}{3}+z^3$$.
Therefore the density is simply the derivative of this, which I found to be $-z^2+3z^2$
Q2: $X$ is the percentage spent on total budget. $Y$ is spend on books. Joint Distribution is $f_2=8xy$ for $0\le x\le 1$ and $0\le y \le x$. Find the marginal densities and sketch both. then find their expected values. Find the conditional density of $f{Y|X}$ and evaluate the $E(Y|X=.6)$. Let $Z=Y/X$. Find the PDF or CDF of $Z$ and find the expected value of $Z$.
Marginal Density of $X$:$\int^x_0 8xydy=4x^3$
Marginal Density of $Y$:$\int^1_y 8xydx=4y-4y^3$
$E(X)=\int^1_0xf_xdx=\int^1_04x^4dx=\frac{4}{5}$
$E(Y)=\int^1_0yf_ydy=\int^1_04y^2-4y^4=\frac{8}{15}$
The conditional density of $f{Y|X}=\frac{8xy}{4x^3}=\frac{4y}{x^2}$
$E(Y|X=.6)=\int^.6_0 y(\frac{4y}{.6^2})=\frac{4}{5}$
Lastly, $$\int^1_0\int^{zx}_08xydydx=\int^1_04xy^2|^{zx}_0=\int^1_04z^2x^3=z^2x^4|^1_0=z^2$$
Thus by taking the derivative the density of $z$ is $2z$.
$E(Z)=\int^1_0 z*2z=\frac{2}{3}$