Function of random variables

**doaa** · May 3rd 2009, 09:12 AM

can some one help in this question

:

If X is Normal( $\mu,$ $\sigma^2$ ) and Y is Exponentioal ( $\lambda$ )
What is the distribution for:

Z=X+Y
Z=X-Y
Z=X*y
Z=x/Y

ineed these distribution quickness

please

**The Second Solution** · May 3rd 2009, 08:39 PM

Originally Posted by doaa

can some one help in this question

:

If X is Normal( $\mu,$ $\sigma^2$ ) and Y is Exponentioal ( $\lambda$ )
What is the distribution for:

Z=X+Y
Z=X-Y
Z=X*y
Z=x/Y

ineed these distribution quickness

please

Are X and Y independent?

**doaa** · May 3rd 2009, 10:19 PM

Yes they independent
please if you can help???

**The Second Solution** · May 5th 2009, 06:02 AM

Originally Posted by doaa

Yes they independent
please if you can help???

Then can't you write down the joint pdf of X and Y and then calculate the cdf of Z in each case.

(You could also use moment generating functions for Z = X + Y but then you'd need to identify the resulting mgf ....)

**matheagle** · May 5th 2009, 05:13 PM

This is a nasty problem. There is no nice answer, i.e., nice form.
IF BOTH X and Y are normal or BOTH are exponential then you can use the MGF. BUT the MGF here won't be recognizable otherwise and that only works for the sum and difference too. So that's worthless. I would just do a 2-2 tranform and integrate out the dummy variable. BUT I wonder if the instructor wanted both of the same form.

**doaa** · May 5th 2009, 11:02 PM

First thank you

But my doctor give me as homework and i tried to solve it but the formulas is complicated
and i though that my work is wrong
and i dont know what to do

**cl85** · May 6th 2009, 12:20 AM

Instead of 2-2 transform, it's probably easier to find $F_Z(z)$ .

i) X + Y
$F_Z(z) = \int_{-\infty}^{\infty} \int_{-\infty}^{z-y}f_X(x)f_Y(y) dx dy$

ii) X-Y
$F_Z(z) = \int_{-\infty}^{\infty} \int_{-\infty}^{z+y}f_X(x)f_Y(y) dx dy$

iii) X*Y
$F_Z(z) = \int_{-\infty}^{\infty} \int_{z/y}^{0}f_X(x)f_Y(y) dx dy + \int_{-\infty}^{\infty} \int_{0}^{z/y}f_X(x)f_Y(y) dx dy$

iv) X/Y
$F_Z(z) = \int_{-\infty}^{\infty} \int_{-\infty}^{yz}f_X(x)f_Y(y) dx dy$

The upper and lower limit might not be correct but I hope you get the idea.

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