Calculating Probability for Continuous Random Vectors

[XodoX]

1. The probability density function for the continuous random vector X = (X₁, X₂) is given by:

Fx₁, x₂(X_1,X2) = { 1/2 if X₁+X₂\leq 2,X₁\geq 0,X₂\geq0
0 otherwise }

Calculate the probability that X₁>2X₂


I do not know which formula to use. I can't find the right one. How do I do this? The joint density formula?

 

[LCKurtz]

Draw a picture in the x₁x₂ plane showing where the density is 1/2. Then, on the same picture, draw the region in question where x₁>2x₂. That should show you the region to work with.

 

[XodoX]

Draw a picture in the x₁x₂ plane showing where the density is 1/2. Then, on the same picture, draw the region in question where x₁>2x₂. That should show you the region to work with.

Yeah, I did that. However, I don't know which formula to use.

 

[Dick]

You integrate the probability density over the region where x1>2*x2.

 

[XodoX]

You integrate the probability density over the region where x1>2*x2.

So joint density was right?

 

[Dick]

So joint density was right?

If that's what you call integrating probability density over part of your region where x1>2*x2, then sure.

 

[XodoX]

Joint density would be:

F x,y(x,y)=xe^-x(y+1), x>0 ,y=0

 

[Dick]

Joint density would be:

F x,y(x,y)=xe^-x(y+1), x>0 ,y=0

That's not it. The region where density is 1/2 is a triangle. Does the picture you drew show that?

 

[XodoX]

That's not it. The region where density is 1/2 is a triangle. Does the picture you drew show that?

Well, I actually didn't know that this one can not be used for triangles?
There's another formula I found for joint density...

F N,X(n,x)=Xⁿe^-2x/n!

 

[Dick]

Well, I actually didn't know that this one can not be used for triangles?
There's another formula I found for joint density...

F N,X(n,x)=Xⁿe^-2x/n!

Stop looking up formulas. You don't need them. It's not that complicated. LCKurtz said to draw a picture. You said you did. What does it look like? Describe the triangle where density is 1/2.

 

[LCKurtz]

Joint density would be:

F x,y(x,y)=xe^-x(y+1), x>0 ,y=0

Huh?? Where did that come from? You gave us the joint density in the original post. And the probability that X₁>2X₂ should be a number between 0 and 1.

 

[XodoX]

Well, I eventually solved it myself. Integration wasn't needed.

 

[Dick]

Well, I eventually solved it myself. Integration wasn't needed.

Good. That's always the best way to solve it. And, yes, you don't need explicit integration since the probability density isn't a function of x1 and x2. You just find the overlap area and multiply by 1/2.