Solutions - 5200-01

(a)

Find the area of the triangle and write the PDF:

The area of the triangle is $\frac{1}{2}$. Therefore the PDF is

$$f_{X,Y}(x,y)=\{\begin{array}{cc}\frac{1}{1/2}=2& \begin{array}{c}{\displaystyle 0\le x\le 1}\\ {\displaystyle 0\le y\le 1-x}\end{array}\\ 0& \text{otherwise}\end{array}$$

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(b)

(1) Integrate with respect to $y$ to find the marginal PDF for $X$:

$$\begin{array}{c}{f}_X(x)={\int }_0^{1-x}2dy\gg \gg {2y|}_0^{1-x}\gg \gg 2(1-x)\end{array}$$ $$f_X(x)=\{\begin{array}{cc}2(1-x)& 0\le x\le 1\\ 0& \text{otherwise}\end{array}$$

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(2) Integrate with respect to $x$ to find the marginal PDF for $Y$:

$$\begin{array}{c}{f}_Y(y)={\int }_0^{1-y}2dx\gg \gg {2x|}_0^{1-y}\gg \gg 2(1-y)\end{array}$$ $$f_Y(y)=\{\begin{array}{cc}2(1-y)& 0\le y\le 1\\ 0& \text{otherwise}\end{array}$$

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(c)

(1) Compute the product $f_X(x)f_Y(y)$ and compare to $f_{X,Y}(x,y)$:

$$2\stackrel{?}{=}4(1-x)(1-y)$$

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(2) Consider the case $x=1,y=0$:

$$4(1-1)(1-0)=0\ne 2$$

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