can some one help in this question:
If X is Normal($\displaystyle \mu,$$\displaystyle \sigma^2$) and Y is Exponentioal ($\displaystyle \lambda$)
What is the distribution for:
Z=X+Y
Z=X-Y
Z=X*y
Z=x/Y
ineed these distribution quickness
please
can some one help in this question:
If X is Normal($\displaystyle \mu,$$\displaystyle \sigma^2$) and Y is Exponentioal ($\displaystyle \lambda$)
What is the distribution for:
Z=X+Y
Z=X-Y
Z=X*y
Z=x/Y
ineed these distribution quickness
please
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.
Instead of 2-2 transform, it's probably easier to find $\displaystyle F_Z(z)$.
i) X + Y
$\displaystyle F_Z(z) = \int_{-\infty}^{\infty} \int_{-\infty}^{z-y}f_X(x)f_Y(y) dx dy$
ii) X-Y
$\displaystyle F_Z(z) = \int_{-\infty}^{\infty} \int_{-\infty}^{z+y}f_X(x)f_Y(y) dx dy$
iii) X*Y
$\displaystyle F_Z(z) = \int_{-\infty}^{\infty} \int_{z/y}^{0}f_X(x)f_Y(y) dx dy + \int_{-\infty}^{\infty} \int_{0}^{z/y}f_X(x)f_Y(y) dx dy$
iv) X/Y
$\displaystyle F_Z(z) = \int_{-\infty}^{\infty} \int_{-\infty}^{yz}f_X(x)f_Y(y) dx dy$
The upper and lower limit might not be correct but I hope you get the idea.