6) Bernoulli and Poisson

6.1 The Bernoulli Process

• Bernoulli Process: A sequence of independent Bernoulli trials, where each trial has two possible outcomes: success (1) with probability $p$, and failure (0) with probability $1-p$.

  • Interarrival Time: The number of trials until the next success follows a geometric distribution with parameter $p$: $$P(X=k)=(1-p{)}^{k-1}p,k=1,2,\dots$$
  • Number of Arrivals: The number of successes in a given number of trials follows a binomial distribution: $$P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k},k=0,1,\dots ,n$$

• Splitting and Merging:

  • Splitting: If a Bernoulli process is split into two by selecting each success independently with probabilities $q$ and $1-q$, the two resulting processes are independent Bernoulli processes with success probabilities $pq$ and $p(1-q)$, respectively.
  • Merging: Two independent Bernoulli processes with success probabilities $p$ and $q$ can be merged into a single Bernoulli process with success probability $p+q-pq$.

6.2 The Poisson Process

• Poisson Process: A continuous-time stochastic process where the number of arrivals in any interval of time follows a Poisson distribution.

  • Arrival Rate: Defined by a parameter $\lambda$ (rate of arrivals per unit of time).
  • Poisson Distribution: The probability of $k$ arrivals in a time interval of length $t$ is: $$P(k,t)=\frac{(\lambda t{)}^k{e}^{-\lambda t}}{k!},k=0,1,2,\dots$$
  • Interarrival Times: The time between consecutive arrivals follows an exponential distribution with rate $\lambda$: $$P(T>t)=e^{-\lambda t},t\ge 0$$

• Splitting and Merging:

  • Splitting: A Poisson process with rate $\lambda$ can be split into two independent Poisson processes with rates $\lambda p$ and $\lambda (1-p)$.
  • Merging: Two independent Poisson processes with rates ${\lambda }_1$ and ${\lambda }_2$ can be merged into a single Poisson process with rate ${\lambda }_1+{\lambda }_2$.

Key Properties of Poisson Processes

• Memorylessness: The time until the next arrival is independent of how much time has already passed.
• Additive Property: The sum of independent Poisson random variables is also a Poisson random variable.