Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let the random point (X, Y ) be uniformly distributed on the triangle $D=\{(x,y):0\leq x\leq y\leq 1\}$ Find the marginal densities of X and Y and plot their graphs.enter image description here

share|improve this question

1 Answer 1

up vote 1 down vote accepted

The joint density function is $2$ in the triangle, and $0$ elsewhere, since the triangle has area $1/2$.

We find the cumulative distribution function $F_Y(y)$ of $Y$. It is not interesting except when $0\le y \le 1$. So suppose that $0\le y\le 1$, and find $P(Y \le y)$. This is the integral of the joint density function over a certain region $R_y$. But we do not need to do much work to evaluate the double integral. But we do need to draw a picture.

The region $R_y$ is an isosceles right triangle whose two legs are equal to $y$. So the area is $y^2/2$, and therefore $F_Y(y)=(2)(1/2)(y^2)=y^2$. It follows by differentiating that the density function $f_Y(y)$ is equal to $2y$ for $0\le y \le 1$. (The density function is $0$ elsewhere.)

For the random variable $X$, the calculation is similar but a little more complicated. The part of the triangle on which $X \le x$ is a trapezoid. Just find its area, multiply by $2$, and differentiate. Alternately, first find $P(X \ge x)$. The geometry is a bit easier.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.