# Conditional probability density function

1. Feb 25, 2009

### drullanorull

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 has a probability density that 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?

2. Feb 25, 2009

$$f_{X|Y}(x|y)=f_Z(x-y)$$