# Thread: A question on joint probability

1. ## A question on joint probability

Hi guys, I have a question about joint probability.

The problem is:
The joint probability fxn for the continuous random variables X and Y is given by
f(x,y) = c, 0<=x<=y<=1 (c is a constant),
= 0 otherwise

Find the probability that X + Y < 1.

I was thinking that I'm supposed to take the double integral, from 0 to 1/2 and from x to 1-x, of (c)dydx, since x < y, but I'm not too sure.

Any help would be greatly appreciated!

2. Originally Posted by probabilitystudent
Hi guys, I have a question about joint probability.

The problem is:
The joint probability fxn for the continuous random variables X and Y is given by
f(x,y) = c, 0<=x<=y<=1 (c is a constant),
= 0 otherwise

Find the probability that X + Y < 1.

I was thinking that I'm supposed to take the double integral, from 0 to 1/2 and from x to 1-x, of (c)dydx, since x < y, but I'm not too sure.

Any help would be greatly appreciated!
Since the joint pdf is a constant, the probability calculation is simple geometry.

Shade the region of the xy-plane defined the support defined by $\displaystyle 0 \leq x \leq y \leq 1$. The area of this region is obviously 1/2.

Now shade the area of this region enclosed by the y-axis and the line x + y = 1/2. Calculate the area A of this region. Then the required probability is clearly $\displaystyle \frac{A}{\frac{1}{2}} = 2A = .....$

3. Sorry, it all makes sense now, but I'm a bit confused about one part. Why is it the line from x+y = 1/2, and not the line x+y < 1

4. Originally Posted by probabilitystudent
Sorry, it all makes sense now, but I'm a bit confused about one part. Why is it the line from x+y = 1/2, and not the line x+y < 1
Misreading on my part. X + Y < 1 it is.

5. It's just one-half of the region 0<x<y<1

or $\displaystyle \int_0^{.5}\int_x^{1-x}2dydx$