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Let $\{N(t) : t \geq 0\}$ be a Poisson process with rate $\lambda\gt 0$. Let $Y$ be a random variable independent of $N(t)$, such that $Y = 1$ with probability $1/2$ and $Y = −1$ with probability $1/2$.

We define the new process $X(t)$ by $X(t) = Y\times(−1)^{N(t)}$. Determine

a) $E[X(t)]\qquad (t \geq 0)$.

b) $Cov(X(s),X(t))\qquad (0 \lt s \lt t \lt\infty)$.

Any Hints?

Can someone please send me a concise reference covering poisson processes that can help in having the intuition required to solve problems like the aforementioned one (topics like intro to poisson processes, non-homogeneous poisson processes, compound poisson processes and ......etc)?

Also I would like to find a good reference for markov chains (up to continuous time MC and M/M/! queues and other related topics).

Mick A
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user220502
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1 Answers1

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Hints:

(a) Conditioning on $Y$ will simplify the problem:

\begin{eqnarray*} E(X(t)) &=& E(X(t)\mid Y=1)P(Y=1) + E(X(t)\mid Y=-1)P(Y=-1) \\ &=& \ldots \end{eqnarray*}

(b)

\begin{eqnarray*} Cov(X(s),X(t)) &=& E(X(s)X(t)) - E(X(s))E(X(t)). \end{eqnarray*}

Use part (a) to evaluate the second term. For the first term, note that

$$X(t) = X(s)(-1)^{N(t-s)}\qquad\qquad\text{(a product of two }\color{black}{independent }\text{ RVs)}.$$

Also, the fact that $(-1)^{N(t-s)}$ is $1$ when $N(t-s)$ is even, otherwise odd, should help you evaluate $E((-1)^{N(t-s)})$ via the usual expectation formula $E(Z)=\sum_z{zP(Z=z)}$.

$\\$

Reference: Introduction to Probability Models by Sheldon Ross covers those topics and is easy-to-read.

Mick A
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