Find the conditional distribution of $X_t$ given $X_t+Y_t=n $ , and find $\Bbb{E}[X_t+Y_t|X_{2t}]$.

Question

Let $\{X_t,t\ge0\}$,$\{Y_t,t\ge0\}$ be independent Poisson process with rates $\lambda$ and $2\lambda $ respectively. Find the conditional distribution of $X_t$ given $X_t+Y_t=n $ , and find $\Bbb{E}[X_t+Y_t|X_{2t}]$.

My try: First, conditional distribution of $X_t$ given $X_t+Y_t=n $. i.e find $P(X_t=k|X_t+Y_t=n)$. We have $P(X_t=k|X_t+Y_t=n)=P(X_t=k,Y_t=n-k)=P(X_t=k)P(Y_t=n-k)$. From this I think all I need do is to make certain what exactly are $P(X_t=k)$ and $P(Y_t=n-k)$, first I think it should go like $P(X_t=k)=e^{-\lambda}\cdot\frac{\lambda^k}{k!}$ and $P(Y_t=n-k)=e^{-2\lambda}\cdot\frac{(2\lambda)^{n-k}}{(n-k)!}$. But I think I lost the parameter $t$. I know $X_t$ is a Poisson process with rate $\lambda$ then $X_t-X_s\sim Poisson(\lambda(t-s))$. How to deduce the distribution of $P(X_t)$ from this. Second , what's the usually step to find $\Bbb{E}[X_t+Y_t|X_{2t}]$.

Newly to this subject without teacher, Thanks in advance.

Answer 1

You know that $X_t+Y_t$ is Poisson distributed with rate $3\lambda t$ , because the intervals over which point events of $X_t$ and $Y_t$ occur are independent , and do so at rates $\lambda t$ and $2\lambda t$ respectively .

If I tell you that a particular point event is one which occurred in one of these intervals, what would you evaluate as the probability that it occurred in the interval of $X_t$ ?

Under condition that $X_t+Y_t=n$, then $X_t$ represent the count of events which have occurred in that interval when given that $n$ events occurred among the two intervals .

Or using your maths...

$$\begin{align}\mathsf P(X_t=k\mid X_t+Y_t=n)&=\dfrac{\mathsf P(X_t=k)\mathsf P(Y_t=n-k)}{\mathsf P(X_t+Y_t=n)}\\[2ex]&=\dfrac{\dfrac{(\lambda t)^k\mathrm e^{-\lambda t}}{k!}\cdot\dfrac{(2\lambda t)^{n-k}\mathrm e^{-2\lambda t}}{(n-k)!}}{\dfrac{(3\lambda t)^n\mathrm e^{-3\lambda t}}{n!}}\quad\big[0\leq k\leq n\big]\\[2ex]&=\dfrac{n!}{k!~(n-k)!}\dfrac{2^{n-k}}{3^n}\quad\big[0\leq k\leq n\big]\end{align}$$

What is the expectation for such a distribution ?

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Similarly to the above, $X_t$ and $X_{2t}-X_t$ count point events occurring in independent Poisson intervals (of length $t$), each with the same average rate, $\lambda t$, so what is $\mathsf E(X_t\mid X_{2t})$?

Finally, $Y_t$ and $X_{2t}$ are independent random variables, so $\mathsf E(Y_t\mid X_{2t})=\mathsf E(Y_t)$.