Bernoulli Process Practice 1

What this lecture covers

This is a worked practice problem, not a formal lecture, applying the Bernoulli process material from earlier in the course. It sets up mosquito bites as a Bernoulli process, finds the mean and variance of the time between bites, then extends the problem to a second insect (ticks) and merges the two bite processes into one.

You should already know Bernoulli processes and geometric random variables before watching. After watching, you'll be able to compute the parameter of a merged Bernoulli process from two independent ones and apply the geometric distribution's mean and variance formulas.

Key ideas

• Bernoulli process parameter: the per-second probability of a bite is the product of the probability an insect lands and the probability it bites given it lands (here 0.5 * 0.2 = 0.1 for mosquitoes).
• Geometric random variable: the time until the first success in a Bernoulli process, with PMF P(x=k) = (1-p)^(k-1) * p.
• Mean and variance of a geometric variable: E[x] = 1/p and Var(x) = (1-p)/p^2.
• Merging independent Bernoulli processes: the merged process succeeds when either component process succeeds, with parameter r = p + q - p*q (equivalently 1 - (1-p)(1-q)).
• Independence assumption: mosquito bites and tick bites are treated as independent Bernoulli processes so their merge formula applies directly.

Before you watch

• Know the definition of a Bernoulli process and its memorylessness.
• Be comfortable with the geometric distribution's PMF, mean, and variance.
• Know how to merge two independent Bernoulli (or Poisson) processes.

Check your understanding

1. Why does the memoryless property let you treat the time to the next bite the same way regardless of past history?
2. How is the merged bite parameter r derived from the mosquito and tick parameters?
3. Why does the variance grow so much faster than the mean as the bite probability shrinks?
4. If a third insect were added with its own independent probability, how would you extend the merging formula?