Friday, April 25, 2014
Bernoulli Process
Time of the \(k^{th}\) success/arrival
| \[ \begin{aligned} Y_k &= \text{ time of }k^{th} \text{ arrival} & \color{blue}{\text{Pascal random variable}} \\ &= T_1 + \cdots + T_k & T_i \text{ are i.i.d, Geometric}(p) \\ T_k &= k^{th}\text{ inter-arrival time } = Y_k - Y_{k-1} \\ \mathbb{E}[Y_k] &= \frac{k}{p} \qquad var(Y_k) = \frac{k(1-p)}{p^2} \\ \mathbb{P}(Y_k = t) &= \mathbb{P}(k-1 \text{ arrivals in }t-1 \text{ time}) \cdot \mathbb{P}(\text{arrival at time } t) \\ p_{Y_k}(t) &= {{t-1} \choose {k-1}} p^k (1-p)^{t-k} \qquad \text{for }t = k, k+1, \cdots & \color{blue}{\text{Pascal PMF}} \text{ of order }k \text{ and parameter }p \\ \end{aligned} \] |
Merging of Bernoulli(\(p\)) and Bernoulli(\(q\))
\(\quad\) Bernoulli(\(p + q - pq\)) \(\qquad\qquad\) Collisions are counted as one arrival
| \[ \begin{aligned} \mathbb{P}(\text{arrival in first process | arrival}) &= \frac{p}{p+q-pq} \\ \end{aligned} \] |
Source: MITx 6.041x, Lecture 21.
