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Probability · Lecture 29 of 76 · 15:10

Mean & Variance of the Exponential

Mean & Variance of the Exponential on YouTube

Study guide

What this lecture covers

This recitation problem works through the exponential distribution in detail: deriving its CDF from its PDF, computing its mean and variance using integration by parts, and then extending to the maximum and minimum of several independent, identically distributed exponential random variables. It builds on the definitions of PDF, CDF, expectation and variance from earlier in the course.

You'll come away able to derive the exponential's CDF, mean and variance from its density, and use the trick of rewriting max <= z and min > w as intersections of simpler events to find the distribution of extreme values of independent random variables.

Key ideas

  • Exponential CDF: for X exponential with rate lambda, F(x) = 1 - e^(-lambda*x) for x >= 0, found by directly integrating the density.
  • Mean via integration by parts: E[X] = 1/lambda, computed by choosing u=t and dv as the density term, with the exponential term dominating the polynomial term in the limit as t -> infinity.
  • Variance from E[X^2]: a similar integration-by-parts calculation gives E[X^2] = 2/lambda^2, so Var(X) = 1/lambda^2.
  • Exponentials always beat polynomials in the limit: any term like t * e^(-lambda*t) or t^2 * e^(-lambda*t) goes to 0 as t goes to infinity, which is why these integrals converge.
  • Max of independent variables: P(max(X1,...,Xn) <= z) equals the product of the individual CDFs, since the max is below z exactly when every variable is below z.
  • Min of independent variables: P(min(X1,...,Xn) > w) equals the product of the individual survival probabilities P(Xi > w), since the min exceeds w exactly when every variable exceeds w.
  • Minimum of exponentials is exponential: the minimum of two independent exponentials with rate lambda is itself exponential, with rate 2*lambda.

Before you watch

  • Know the exponential distribution's PDF and the definitions of CDF, expectation and variance.
  • Be comfortable with integration by parts and with taking limits of products like t * e^(-t) as t -> infinity.
  • This problem assumes you've seen continuous random variables and independence from earlier lectures in the course.

Check your understanding

  1. Why does the derivation choose u=t rather than u = e^(-lambda*t) when applying integration by parts for the mean?
  2. Why does P(max(X1,X2,X3) <= z) factor into a product of individual CDFs, but P(min(X1,X2) <= w) cannot be handled the same way directly?
  3. Why is the minimum of two independent rate-lambda exponentials itself exponential, and what is its rate?
  4. How would the formula for the PDF of the maximum change with four independent exponentials instead of three?

Vocabulary

exponential distribution (noun)
A distribution used for waiting times, where events happen at a constant average rate.
The exponential distribution models the time until the next bus arrives.
rate (noun)
How often something happens per unit of time.
Lambda is the rate parameter of the exponential distribution.
derive (verb)
To work out a formula step by step from known rules.
We derive the CDF by integrating the density.
integration by parts (phrase)
A calculus technique for integrating a product of two functions.
Integration by parts is used to compute the mean of the exponential.
mean (noun)
The average value of a random variable.
The mean of the exponential distribution is 1 over lambda.
variance (noun)
A number that measures how spread out a random variable's values are.
The variance of the exponential is 1 over lambda squared.
dominate (verb)
To become much larger or more important than something else.
The exponential term dominates the polynomial term as t grows.
converge (verb)
To settle toward a fixed value as a process continues.
The integral converges because the exponential decays fast enough.
limit (noun)
The value a quantity approaches as another quantity grows very large or small.
We take the limit as t goes to infinity.
intersection (noun)
The set of outcomes that satisfy two or more conditions at once.
The event 'max is below z' is an intersection of simpler events.
identically distributed (phrase)
Having exactly the same probability pattern as other variables.
We consider n independent, identically distributed exponentials.
survival probability (phrase)
The probability that a random variable is still above a certain value.
The survival probability of an exponential decays as time passes.
extreme value (phrase)
The largest or smallest value among a group of outcomes.
We study the extreme values, the max and min, of several exponentials.
probability density function (PDF) (phrase)
A function describing how likely a continuous variable is near each value.
The exponential's PDF is integrated to find its CDF.
cumulative distribution function (CDF) (phrase)
A function giving the probability that a variable is at or below a value.
The exponential CDF is 1 minus e to the negative lambda x.
waiting time (phrase)
The amount of time until an event happens.
The exponential distribution often models a waiting time.
polynomial (noun)
An expression built from powers of a variable, like t or t squared.
The exponential term always beats a polynomial term as t grows.
decay (verb)
To gradually shrink toward zero.
The survival probability decays as time passes.
product (noun)
The result of multiplying two or more numbers or probabilities together.
The max's CDF equals the product of the individual CDFs.
exceed (verb)
To be greater than a certain value.
The min exceeds w exactly when every variable exceeds w.
extend (verb)
To apply an idea further, to a larger or new case.
The problem extends the exponential's properties to the max and min of several.
trick (noun)
A clever, non-obvious method for solving a problem.
Rewriting max and min as intersections is the key trick here.
exactly (adverb)
Precisely and with no difference at all.
The max is below z exactly when every variable is below z.
calculus (noun)
The branch of mathematics dealing with rates of change and integration.
Integration by parts is a standard calculus technique used here.
come away (phrasal verb)
To finish an activity having gained a certain skill or understanding.
You should come away able to derive the exponential's mean and variance.

Chapters

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: Jagdish Ramakrishnan

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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