# Conditional probability density function

• February 25th 2009, 12:11 AM
Robert Hall
Conditional probability density function
Please help me with this. Any suggestions are greatly appreciated.
Imagine that I have a bank account. X is the amount of cash on the account at time t+1. Y is the amount of cash at time t. The amount of cash depends on the deposits made and on the amount of cash during the previous period. The deposits are made based on a random variable, Z, (stock returns) which is log-normal distributed. My question is what the probability density function for the amount of cash at time t+1 looks like.
$f_X(x)=\int f_{X,Y}(x,y)dy = \int f_{X|Y}(x|y)f_Y(y)dy$
My problem is how to relate $f_{X|Y}(x|y)$ with the deposits.
Is $f_{X|Y}(x|y)=y+f_Z(z)$
So that
$f_X(x)=\int(y+f_Z(z))f_Y(y)dy$

Is this true?
• February 28th 2009, 08:52 PM
meymathis
Obviously if Y and Z are independent this is easy since the probability density function of a sum is the convolution of the summands' probability densities. So you get $f_X(x) = f_Y*f_Z(x)$.

I.e. $f_X(x) = \int_{-\infty}^\infty f_Y(x-t) f_Z(t)\,dt$.

You can't really do this if they are dependent without knowing more information. Namely Y and Z's joint density function.