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beyondlight

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## 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**

[tex]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)}[/tex]

Next step is following, lets call this part 2:

[tex]\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}}[/tex]

This is further simplified (part 3):

[tex]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][/tex]

**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?