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Probability · Lecture 51 of 76 · 8:21

Bernoulli Process Practice 1

Study guide

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

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

Vocabulary

mosquito bite (phrase): A bite from a small flying insect, used here as an example of a random arrival.
Each mosquito bite is treated as an arrival in a Bernoulli process.
land (verb): To come down and settle on a surface.
The probability an insect lands is part of the bite process.
per-second probability (phrase): The chance of an event happening within any single second.
The per-second probability of a bite is 0.1.
component process (phrase): One of the smaller processes that are combined into a larger one.
Mosquito bites are one component process in the merged process.
shrink (verb): To become smaller.
As the bite probability shrinks, the variance grows much faster than the mean.
Bernoulli process (phrase): A sequence of independent trials, each succeeding with the same fixed probability.
Mosquito bites are modeled as a Bernoulli process.
geometric random variable (phrase): A random variable counting the number of trials until the first success.
The time until the first bite is a geometric random variable.
mean (noun): The average value of a random variable.
The mean time between bites is 1/p.
variance (noun): A number that measures how spread out a random variable's values are.
The variance of the geometric variable is (1-p)/p^2.
merge (verb): To combine two things into one.
We merge the mosquito and tick bite processes into one.
merged process (phrase): The single process formed by combining two or more independent processes.
The merged process succeeds whenever either insect bites.
parameter (noun): A fixed number that controls the behavior of a distribution or process.
The merged process has its own success parameter r.
independence (noun): The property of not being affected by another event or variable.
Independence between the two bite processes lets us apply the merge formula.
formula (noun): A fixed rule written with symbols that gives a result from given values.
The merge formula gives r = p + q - p*q.
equivalently (adverb): In another way that means exactly the same thing.
Equivalently, r can be written as 1 - (1-p)(1-q).
insect (noun): A small creature such as a mosquito or tick.
A second insect, the tick, is added to the problem.
extend (verb): To make an idea apply to a larger or new situation.
The problem extends to a second insect and process merging.
practice problem (phrase): An exercise used to apply and test understanding of a method.
This is a practice problem, not a formal lecture.
faster (adverb): At a greater speed or rate than something else.
The variance grows faster than the mean as p shrinks.
assumption (noun): Something taken to be true without being directly proven.
Independence between the two bite processes is a key assumption.
apply (verb): To use a rule or method in a specific situation.
We apply the geometric distribution's mean and variance formulas.
derive (verb): To work out a result step by step from known rules or facts.
We derive the merged bite parameter r from the two insect probabilities.
given (that) (phrase): Assuming a certain fact or condition is true.
Given that an insect lands, it may or may not bite.
treat (verb): To consider or deal with something in a particular way.
We treat mosquito and tick bites as independent Bernoulli processes.
setup (noun): The starting conditions and definitions of a problem.
The setup gives the per-second probability of a mosquito bite as 0.1.

From the YouTube description

MIT 6.041SC Probabilistic Systems Analysis and Applied Probability, Fall 2013
View the complete course: http://ocw.mit.edu/6-041SCF13
Instructor: Qing He

License: Creative Commons BY-NC-SA
More information at http://ocw.mit.edu/terms
More courses at http://ocw.mit.edu

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