How to solve poisson process probabilities

[beyondlight]

Homework Statement

Let X(t) and Y(t) be independent Poisson processes, both with rates. Define Z(t)=X(t)+Y(t).
Find E[X(1)|Z(2)=2].



2. The attempt at a solution

$$P(X(1)=k|Z(2)=2)=P(X(1)=k|X(2)+Y(2)=2)=\frac{P(X(1)=k,X(2)+Y(2)=2)}{P(X(2)+Y(2)=2)}=\\\frac{P(X(1)=k,X(2)+Y(2)=2,Y(2)=0)+P(X(1)=k,X(2)+Y(2)=2,Y(2)=1)+P(X(1)=k,X(2)+Y(2)=2,Y(2)=2)}{P(X(2)=2,Y(2)=0)+P(X(2)=1,Y(2)=1)+P(X(2)=0,Y(2)=2)}$$

Next step is following, let's call this part 2:

$$\frac{P(X(1)=k)P(X(2)-X(1)=2-k)P(Y(2)=0)+P(X(1)=K)P(X(2)-X(1)=1-k)P(Y(2)=1)+P(X(1)=k)P(X(2)-X(1)=-k)P(Y(2)=2)}{0.5e^{-2}+e^{-2}+0.5e^{-2}}$$

This is further simplified (part 3):

$$0.5e[P(X(1)=k)P(X(1)=2-k)+P(X(1)=k)P(X(1)=1-k)+P(X(1)=k)P(X(1)=-k)0.5]$$

3. Basic theory

I know that two non-intersecting intervals are independent and that is why the expression X(2)-X(1) is desirable.

Bayes rule and the rule of independence are used in these equations.

However i do not understand the steps in the equations at all...Can someone explain how to think when solving this equation?

 

[Ray Vickson]

Homework Statement

Let X(t) and Y(t) be independent Poisson processes, both with rates. Define Z(t)=X(t)+Y(t).
Find E[X(1)|Z(2)=2].



2. The attempt at a solution

$$P(X(1)=k|Z(2)=2)=P(X(1)=k|X(2)+Y(2)=2)=\frac{P(X(1)=k,X(2)+Y(2)=2)}{P(X(2)+Y(2)=2)}=\\\frac{P(X(1)=k,X(2)+Y(2)=2,Y(2)=0)+P(X(1)=k,X(2)+Y(2)=2,Y(2)=1)+P(X(1)=k,X(2)+Y(2)=2,Y(2)=2)}{P(X(2)=2,Y(2)=0)+P(X(2)=1,Y(2)=1)+P(X(2)=0,Y(2)=2)}$$

Next step is following, let's call this part 2:

$$\frac{P(X(1)=k)P(X(2)-X(1)=2-k)P(Y(2)=0)+P(X(1)=K)P(X(2)-X(1)=1-k)P(Y(2)=1)+P(X(1)=k)P(X(2)-X(1)=-k)P(Y(2)=2)}{0.5e^{-2}+e^{-2}+0.5e^{-2}}$$

This is further simplified (part 3):

$$0.5e[P(X(1)=k)P(X(1)=2-k)+P(X(1)=k)P(X(1)=1-k)+P(X(1)=k)P(X(1)=-k)0.5]$$

3. Basic theory

I know that two non-intersecting intervals are independent and that is why the expression X(2)-X(1) is desirable.

Bayes rule and the rule of independence are used in these equations.

However i do not understand the steps in the equations at all...Can someone explain how to think when solving this equation?

1) How can you be writing down equations that you do not understand? Are you copying them from somewhere?
2) What are the rates? You never told us that, and it certainly matters a lot.
3) Events like X(2)-X(1)=-k cannot occur if k ≥ 1---why not?
4) Where do the 0.5 factors come from? I think I can guess, but should not have to.
5) So, finally, what do the expressions simplify to? You need to figure that out in order to compute the conditional expectation.

 

[beyondlight]

1) i wrote them down from the answers.
2) rate is 1 for both variables
3)?
4)i plugg in the value k for either X or Z in the PMF for the Poisson random variable:
f=(1^ke^(-1))/k!

5) Right now I don't need help with the final answer, I just want to know how you should reason in order to get the expressions that i wrote down.

 

[Ray Vickson]

1) i wrote them down from the answers.
2) rate is 1 for both variables
3)?
4)i plugg in the value k for either X or Z in the PMF for the Poisson random variable:
f=(1^ke^(-1))/k!

5) Right now I don't need help with the final answer, I just want to know how you should reason in order to get the expressions that i wrote down.

You shouldn't, since the expressions you wrote down are wrong (at least, some of them are). Start over.