Let $\{X_t\},\{Y_y\},\{Z_t\}$ be independent Poisson processes, of rates $\lambda_1,~\lambda_2\,\lambda_3,$ respectively. Assume that events arrive from all the above processes. Find the probability the next two events (from now) concern the process $\{X_t\}.$

Attempt. Let $t>0$ be the time we get exactly two events, that is $X_t+Y_t+Z_t=2$, where $X_t,~Y_t,~Z_t$ are rv's from the Poisson distribution of parameters $\lambda_1t,~\lambda_2t,\,\lambda_3t,$ respectively. Given $X_t+Y_t+Z_t=2$, the rv $X_t$ is from the Binomial distribution $Bin\Big(2,\frac{\lambda_1}{\lambda_1+ \lambda_2+\lambda_3}\Big)$ (classic result from probability), so the desired probability of $X_t=2$ is $\Big(\frac{\lambda_1}{\lambda_1+ \lambda_2+\lambda_3}\Big)^2.$

Thanks in advance for the tip!


Good job. The answer is indeed

$$\left( \frac{\lambda_1}{\sum_{i=1}^3\lambda_i}\right)^2$$

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.