# Thread: A question related to Derived Distributions

1. ## A question related to Derived Distributions

Can anyoner help me to solve this question... I'm really stuck on this one

Andy is vacationing in Las Vegas. The amount X (in Dolars) he takes to casino each evening is a random variable
with a PDF of the form
fX(x)=ax if 0≤x≤23 and
0 otherwise.
At the end of each night, the amount Y that he has when leaving the casino is uniformly distributed between
zero and twice the amount that he came with.
Determine the joint PDF fX,Y(x,y).
fX,Y(8,14)=
What is the probability that on a given night Andy makes a positive profit at the casino?
Find the PDF of Andy's profit Z=Y-X on a particular night.
fZ(0)=
Determine the expected value of Z=Y-X.
E[Z]=

Thank you very much.

2. I'm not sure I follow all of this.
Parts seem hazy, especially dollars being continuous.
Here's what I get...

$\displaystyle f_X(x)={x\over 264.5}$ on (0,23)

Next you have Y given X as uniform on 2X....

So $\displaystyle f_{X,Y}(x,y)=f_X(x)f_{Y|X}(x,y)= \left({x\over 264.5}\right)\left({1\over 2x}\right)$

$\displaystyle ={1\over 529}$ where we have 0<x<23 and 0<y<2x

which I would rewrite as either 0<y<2x<46 or 0<y/2<x<23

IS this what you're looking for?
I can get the density of Z=Y-X if you wish, but I would only do that if this makes any sense to you.

3. If you want the probability Y>X, then you need to integrate over that region.
But since the joint density is constant you can use geometry to find that volume.

4. In order to obtain the density of Z=Y-X, you can first obtain the cummulative of Z, then differentiate.

$\displaystyle P(Z\le z)=P(Y-X\le z)$

Now draw the triangle where X,Y exists, that's inside the region, 0<x<23, 0<y<46 and y=2x.
To integrate, but as I said before you can use geometry since the joint density is constant, 1/529,
you find the probabiity of Y<X+z, for any z.
First you draw y=x+z and then figure out where you need to integrate.

5. I got fX(x) and fX,Y(x,y) the same way you did, matheagle.
and then I tried to do this FZ(z) = P(Y-X≤z) = ∫∫ {X,Y|Y-X≤z} fX,Y(x,y) dx dy.
But couldn't figure it out.

For you method, isn't gonna be like ∫(2x-x)dx?
Pardon me if I'm wrong, my calculus is rusty at best.

6. There is no need for calculus, the density is constant, use geometry.
You must always draw your region in the xy-plane.

7. Got it!! T__T
Dude, you're an angel.

8. What if I want to use calculus? Can you give me a hint?