Poisson Process Conditional Distribution

[jiml]

Homework Statement

$X_t$ and $Y_t$ are poisson processes with rates $a$ and $b$

$n = 1,2,3...$Find the CDF $F_X{}_t{}_|{}_X{}_t{}_+{}_Y{}_t{}_={}_n(x)$

Homework Equations

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.

 

[Ray Vickson]

Homework Statement

$X_t$ and $Y_t$ are poisson processes with rates $a$ and $b$

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


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

Homework Equations

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.

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
$$F_{X_t|X_t + Y_t = n}(m) = P(X_t \leq m|X_t+Y_t = n).$$

 

[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

 

[Ray Vickson]

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

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

 

[jiml]

ok,thanks