A question related to Derived Distributions

[essedra]

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.

 

[matheagle]

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.

 

[matheagle]

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.

 

[matheagle]

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.

 

[primus]

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.

 

[matheagle]

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

 

[primus]

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

 

[primus]

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