# Thread: Conditional probability density function

1. ## Conditional probability density function

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.
$\displaystyle f_X(x)=\int f_{X,Y}(x,y)dy = \int f_{X|Y}(x|y)f_Y(y)dy$
My problem is how to relate $\displaystyle f_{X|Y}(x|y)$ with the deposits.
Is $\displaystyle f_{X|Y}(x|y)=y+f_Z(z)$
So that
$\displaystyle f_X(x)=\int(y+f_Z(z))f_Y(y)dy$

Is this true?

2. 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 $\displaystyle f_X(x) = f_Y*f_Z(x)$.

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

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