# Poisson Process Conditional Distribution

1. Apr 4, 2013

### jiml

1. The problem statement, all variables and given/known data
$X_t$ and $Y_t$ are poisson processes with rates $a$ and $b$

$n = 1,2,3...$

Find the CDF $F_X{}_t{}_|{}_X{}_t{}_+{}_Y{}_t{}_={}_n(x)$

2. Relevant equations

3. The attempt at a solution
$F_X{}_t{}_|{}_X{}_t{}_+{}_Y{}_t{}_={}_n(x)$

$=P(X_t<x|X_t+Y_t=n)$

$=\frac{P(X_t<x,X_t+Y_t=n)}{P(X_t+Y_t=n)}$

Not sure from here, but here goes:

$=\frac{P(Y_t>n-x)}{P(X_t+Y_t=n)}$

$=1-\frac{P(Y_t<=n-x)}{P(X_t+Y_t=n)}$

Not sure if doing correctly.

2. Apr 4, 2013

### Ray Vickson

Since X and Y are counting processes, you should probably avoid using the letter 'x' for values of them, so instead, should use something like $F_{X_t|X_t + Y_t = n}(m).$ Note also that the standard definition of a cdf involves '≤', not '<', so
$$F_{X_t|X_t + Y_t = n}(m) = P(X_t \leq m|X_t+Y_t = n).$$

3. Apr 4, 2013

### jiml

Can someone please help me with my solution, whether I am on the right track in my steps to get to a solution. Thanks

4. Apr 5, 2013

### Ray Vickson

All you have done is use the definition of conditional probability; you are nowhere near the final solution.

5. Apr 5, 2013

### jiml

ok,thanks

Last edited: Apr 6, 2013