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Probability · Lecture 28 of 76 · 13:24

A Mixed Distribution Example

Study guide

What this lecture covers

This recitation problem works through a mixed random variable: Al's waiting time for a taxi or bus, which is exactly zero with some probability, exactly five minutes with some probability, and continuously distributed in between otherwise. It builds directly on the earlier lecture that introduced mixed random variables and the total expectation and total probability theorems.

You'll see how to organize a mixed-variable problem using a partition of disjoint events, how the total expectation theorem extends across discrete and continuous branches at once, and how to build a piecewise CDF by working out each range of x separately.

Key ideas

Partitioning with a tree: the outcome splits into three disjoint, exhaustive events (catching a waiting taxi, catching the next taxi within 5 minutes, or catching the bus at 5 minutes), which together form a partition.
Conditional density from a uniform: conditioning the taxi's arrival time on "it arrives within 5 minutes" rescales the original uniform density over [0,10] to a uniform density of height 1/5 over [0,5].
A mixed random variable takes a discrete value with positive probability on some branches (X=0 or X=5) and a continuous value on another branch, so it can't be described by a PMF or a PDF alone.
Total expectation theorem across mixed branches: E[X] = P(B1)E[X|B1] + P(B2)E[X|B2] + P(B3)E[X|B3] works the same way whether each branch is discrete or continuous.
Total probability theorem for the CDF: P(X <= x) is computed by summing P(Bi) * P(X <= x | Bi) over the partition, for each range of x.
Building a CDF piecewise: the CDF is 0 below the variable's minimum possible value, 1 above its maximum, and requires separate reasoning for the range in between, where contributions from different branches can behave differently (some being all-or-nothing, others being areas under a density).

Before you watch

Review the earlier lecture on mixed random variables and how their CDFs combine jumps and continuous growth.
Know the total expectation theorem and total probability theorem from conditioning on a partition of events.
Be comfortable with the uniform distribution and how conditioning changes its density.

Check your understanding

Why is Al's waiting time a mixed random variable rather than purely discrete or purely continuous?
Why does conditioning the taxi arrival time on "arrives within 5 minutes" produce a uniform density of height 1/5 rather than 1/10?
How does the total expectation theorem apply here even though the branches have different types of distributions?
Why is the probability contribution from the bus branch (B3) exactly 0 for the CDF's middle range, but exactly 1 for x >= 5?

Vocabulary

mixed random variable (phrase): A variable that has both discrete jumps and continuous spread in its probability.
Al's waiting time is a mixed random variable.
partition (noun): A set of events that cover every possibility without overlapping.
The outcomes split into a partition of three disjoint events.
disjoint (adjective): Not overlapping; unable to happen at the same time.
The three branches of the tree are disjoint events.
exhaustive (adjective): Covering every possible outcome, with nothing left out.
The partition is exhaustive because one of the branches must happen.
branch (noun): One separate path or case in a tree of possible outcomes.
Each branch of the tree leads to a different waiting time.
rescale (verb): To change the size or scale of something while keeping its shape.
Conditioning rescales the uniform density to a smaller interval.
uniform distribution (noun): A distribution where every value in a range is equally likely.
The taxi's arrival time follows a uniform distribution.
total expectation theorem (phrase): A rule for finding an overall average by combining averages from each branch of a partition.
The total expectation theorem combines the three branches' averages.
total probability theorem (phrase): A rule for finding an overall probability by combining probabilities from each branch of a partition.
The total probability theorem builds the CDF from the three branches.
condition on (phrase): To assume a certain event is true and calculate probabilities within that assumption.
We condition on the taxi arriving within 5 minutes.
minimum (noun): The smallest possible value.
The CDF is 0 below the variable's minimum possible value.
maximum (noun): The largest possible value.
The CDF is 1 above the variable's maximum possible value.
all-or-nothing (adjective): Having only two possible outcomes, complete or none at all.
The bus branch is all-or-nothing: either it happens or it doesn't.
probability density function (PDF) (phrase): A function describing how likely a continuous variable is near each value.
A mixed variable can't be described by a PDF alone.
probability mass function (PMF) (phrase): A function giving the probability of each possible value of a discrete variable.
A mixed variable can't be described by a PMF alone either.
cumulative distribution function (CDF) (phrase): A function giving the probability that a variable is at or below a value.
The CDF is built piecewise across the three branches.
continuous (adjective): Able to take any value within a range, without gaps.
The taxi's waiting time is continuous between 0 and 5 minutes.
discrete (adjective): Taking only separate, distinct values.
X=0 and X=5 are the discrete parts of the mixed variable.
jump (noun): A sudden increase in a function's value at one specific point.
A mixed variable's CDF combines jumps and continuous growth.
piecewise (adjective): Defined by different rules or formulas over different ranges.
The CDF is built piecewise, one range of x at a time.
arrival (noun): The moment something shows up or happens.
The taxi's arrival time is uniformly distributed.
catch (verb): To successfully get on or use a vehicle before it leaves.
Al might catch a waiting taxi immediately.
reasoning (noun): A logical process of thinking through a problem step by step.
The middle range of the CDF requires separate reasoning for each branch.
contribution (noun): The part that one piece adds to a total result.
Each branch's contribution to the CDF is added separately.
organize (verb): To arrange information in a clear, structured way.
The problem is organized using a partition of disjoint events.
tree (noun): A diagram showing how an outcome branches into different possible paths.
A tree separates the outcome into three disjoint events.

Chapters

0:00 Introduction
0:10 Problem Statement
0:51 Tree Structure
4:12 Expected Value
7:14 CDF

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
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