I would really appreciate if you could help out with the following question

Let $W_1,W_2,\ldots$ be the event times in a Poisson process $\{X(t);t\ge0\}$ of rate $\lambda$, and let $f(w)$ be an arbitrary function. Verify that $$ \mathbb{E}\left[\sum_{i=1}^{X(t)}f(W_i)\right]=\lambda\int_0^tf(w)\,dw. $$

(originally posted to this link: http://i.stack.imgur.com/3e6mf.png). I tried conditioning on X(t) = n but I am unsure how to proceed after that... Also I am not exactly sure how to show this is true since f(w) is an arbitrary function and I do not understand what that means.

Thanks a bunch for the help! I really

  • $\begingroup$ Why don't you write out the question rather than just linking to it? $\endgroup$ Mar 18 '11 at 1:01
  • $\begingroup$ I edited the question to add this. $\endgroup$ Mar 18 '11 at 1:11
  • $\begingroup$ PASTA: "Poisson Arrivals See Time Averages" $\endgroup$ Mar 18 '11 at 1:34
  • $\begingroup$ @George i didn't know how to put math symbols on this forum, so I attached the image. but thanks for your response and thanks for transcribing it here. $\endgroup$
    – icobes
    Mar 18 '11 at 3:59

Hint: Indeed, condition on $X(t)=n$. Then, note that given $X(t)=n$, the locations of the $n$ points are distributed as $n$ i.i.d. uniform$[0,t]$ rv's. This should lead you easily to the result, noting that ${\rm E}f(U)=\frac{1}{t}\int_0^t {f(w)dw} $, where $U$ is a uniform$[0,t]$ rv.

EDIT: More precisely (cf. cardinal's comment below), given $X(t)=n$, the points $W_1,\ldots,W_n$ are distributed as $n$ order statistics from the uniform distribution on $[0,t]$. However, the sum $f(W_1)+\cdots+f(W_n)$ (given $X(t)=n$) is equal in distribution to the sum $f(U_1)+\cdots+f(U_n)$ where the $U_i$ are i.i.d. uniform$[0,t]$ rv's, and thus the result follows straightforwardly using the hint.

EDIT (additional hint, in response to the OP's request): $$ \sum\nolimits_{n = 0}^\infty {n{\rm P}(X(t) = n)} = {\rm E}[X(t)] = ? $$ The left-hand side is given by $$ \sum\nolimits_{n = 0}^\infty {n{\rm E}[f(U)]{\rm P}(X(t) = n)}, $$ where $U$ is a uniform$[0,t]$ random variable.

EDIT (in response to the OP's request): In general, if $Y$ is a random variable with density function $h$, then ${\rm E}[f(Y)] = \int {f(y)h(y)dy} $. A uniform$[0,t]$ random variable, $U$, has constant density function $h(y)=1/t$ for $y \in [0,t]$ (and $0$ otherwise). From this it follows that ${\rm E}[f(U)] = \frac{1}{t}\int_0^t {f(y)dy} $.

As for the summation $\sum\nolimits_{n = 0}^\infty {n{\rm E}[f(U)]{\rm P}(X(t) = n)}$, first note that $$ {\rm E}\Big[\sum\nolimits_{i = 1}^{X(t)} {f(W_i )} \Big] = \sum\limits_{n = 0}^\infty {{\rm E}\Big[} \sum\nolimits_{i = 1}^n {f(W_i )} |X(t) = n \Big]{\rm P}(X(t) = n). $$ Now, recall that given $X(t)=n$, the sum $f(W_1)+\cdots+f(W_n)$ is equal in distribution to the sum $f(U_1)+\cdots+f(U_n)$, where the $U_i$ are i.i.d. uniform$[0,t]$ rv's. This accounts for the $n {\rm E}[f(U)]$ in the sum.

Further questions?

  • $\begingroup$ They are i.i.d. under a permutation of the indices of the $W_i$, but that's an admittedly minor point. $\endgroup$
    – cardinal
    Mar 17 '11 at 22:37
  • $\begingroup$ @cardinal: I'll edit the answer to make that point more precise. $\endgroup$
    – Shai Covo
    Mar 17 '11 at 22:41
  • $\begingroup$ Hi, I am stuck on this part: Sum from n = 0 to infinite of E[sum of i = 1 to n of f(Ui)] * P(X(t) = n). I find the expectation which is n/t and when multiplied by P(X(t) = n) that equals to 1/t * E[X(t)] but I am not sure what to do after... Thanks! $\endgroup$
    – icobes
    Mar 17 '11 at 23:38
  • $\begingroup$ @icobes: Note that $\sum\nolimits_{n = 0}^\infty {nP(X(t) = n)} = E[X(t)]$. Need more hints? $\endgroup$
    – Shai Covo
    Mar 17 '11 at 23:51
  • $\begingroup$ Yep, so I know that 1/t * E[X(t)] is the LHS. I also know that the expected value of a Poisson RV is lambda which makes it lambda * 1/t. But I have no clue how to make that equal to the RHS. A few more hints would be great! Thank you :) $\endgroup$
    – icobes
    Mar 17 '11 at 23:56

One way of solving this problem is to rewrite the sum as a Riemann-Stieltjes integral, $$ \sum_{i=1}^{X(t)}f(W_i)=\int_0^tf(s)\,dX_s. $$ Using the fact that $X(s)$ has expectation $\mathbb{E}[X(s)]=\lambda s$, you can take expectations* of this to get $$ \mathbb{E}\left[\sum_{i=1}^{X(t)}f(W_i)\right]=\lambda\int_0^tf(s)ds. $$

(*) To be rigorous, you should show that the expectation does indeed commute with the integral in this way. However, this is a general result which can be used in lots of situations, and is a very special case of the fact that stochastic integrals with respect to martingales are martingales ($X(s)-\lambda s$ is actually a martingale). In the case where $f$ is piecewise linear, the integral reduces to a sum, and expectations commute with summation by linearity. The fact that it extends to the general case of locally integrable $f$ is a consequence of the monotone class theorem.


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