A Mixed Distribution Example

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

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