← Stat 110: Probability

Unit 9: Order Statistics and Conditional Expectation

Sorting a sample and asking what its extremes look like, then the deepest tool in the course — conditioning a random variable on another random variable.

9 topics · 32 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

9.1 Order Statistics

  1. How it worksHow does conditioning on which value is smallest, middle, or largest build each order statistic's distribution?
  2. Common mistakeWhy isn't the median of a sample the same as the average of the sample's largest and smallest values?
  3. Worked exampleFind the CDF of the maximum of 3 independent Uniform(0,1) random variables at x=0.8.
  4. Going furtherWhat happens to the distribution of the maximum of n Uniform(0,1) draws as n grows very large?
Your turn1 of 2
Find the CDF of the maximum of 8 independent Uniform(0,1) random variables at x = 0.654.

9.1 The Distribution of the Minimum

  1. How it worksHow does 1 minus the product of survival probabilities give the CDF of the minimum?
  2. Worked exampleFind the probability the minimum of 4 independent Exponential(1) random variables exceeds 2.
  3. Going furtherWhat happens to the minimum of n independent Exponential random variables as n grows large?
Your turn1 of 2
Find the probability the minimum of 11 independent Exponential(1) random variables exceeds 1.98.

9.2 Conditional Expectation Given a Random Variable

  1. How it worksHow does averaging Y over its conditional distribution at each value of X build the function E[Y given X]?
  2. Worked exampleFind E[Y given X=2] when Y given X=x is distributed Binomial(10, x/10).
  3. Common mistakeWhy isn't E[Y given X] the same thing as E[Y given X=x] evaluated at one particular number?
  4. Ties togetherHow does E[Y given X] connect back to the ordinary conditional expectation used for events?
Your turn1 of 2
Y given X=x is distributed Binomial(10, x/10). Find Var(Y given X = 7).

9.2 Taking Out What's Known

  1. How it worksHow does treating X as a known constant, once you're conditioning on it, justify pulling g(X) out front?
  2. Worked exampleFind E[X times Y given X] when E[Y given X] equals 3X, by taking X out of the expectation.
  3. Common mistakeWhy doesn't taking out what's known work for a function of Y when you're conditioning on X?
Your turn1 of 2
E[Y given X] equals 6X. Find E[X times Y given X] when X = 7, by taking X out of the expectation.

9.3 Adam's Law (The Tower Property)

  1. How it worksHow does the tower property let you compute a hard expectation in two easier conditional steps?
  2. Common mistakeWhy isn't E[E[Y given X]] just a needlessly complicated way to write E[Y]?
  3. Worked exampleFind E[Y] when Y given X is Binomial(10,X) and X is uniform on the interval from 0 to 1.
  4. Going furtherWhat happens to Adam's law when you condition on two random variables instead of one?
Your turn1 of 2
Y given X is Binomial(36, X), and X is uniform on 0 to 1. Find E[Y] using Adam's law.

9.3 Eve's Law (The Law of Total Variance)

  1. How it worksHow does Eve's law combine the average conditional variance with the variance of the conditional mean?
  2. Worked exampleFind Var(Y) when E[Y given X] equals 3X, Var(Y given X) equals 2, and X is uniform on 0 to 1.
  3. Common mistakeWhy doesn't the variance of Y given X alone capture all of Y's total variability?
  4. Ties togetherHow does Eve's law connect to Simpson's paradox once a population is split into subgroups?
Your turn1 of 2
E[Y given X] equals 4X, Var(Y given X) equals 3.42, and X is uniform on 0 to 1 (so Var(X) = 1/12). Find Var(Y) using Eve's law.

9.4 The Two Envelope Paradox

  1. How it worksHow does an unstated assumption about the amounts inside the envelopes create the paradox's contradiction?
  2. Worked exampleFind the flaw in claiming switching envelopes gains 1.25 times the current envelope's amount for every amount x.
  3. Common mistakeWhy doesn't a proper prior distribution on the envelope amounts actually favor switching?
  4. Going furtherWhat happens to the two-envelope paradox when the smaller amount is drawn from a known, valid prior?
Your turn1 of 2
Two envelopes contain 38 dollars and 118 dollars respectively, and you're equally likely to be holding either one. Find the expected value of the envelope you end up with if you always switch.

9.4 Waiting Time for HT Versus HH

  1. How it worksHow does conditioning on the first flip and using Adam's law solve for each pattern's expected wait?
  2. Worked exampleFind the expected number of flips to see HT and the expected number to see HH with a fair coin.
  3. Common mistakeWhy doesn't HH need the same expected waiting time as HT just because both are two-flip patterns?
Your turn1 of 2
A coin lands heads with probability 0.618 on each flip. Find the expected number of flips to see the pattern HT (heads then tails).

9.5 The Sum of a Random Number of Random Variables

  1. How it worksHow does conditioning on the random count N reduce the sum's mean and variance to formulas built from N?
  2. Worked exampleFind E[total claims] when the number of claims N is Poisson with mean 20 and each claim averages 500 dollars.
  3. Going furtherWhat happens to the variance of a random sum when the number of terms N is itself random rather than fixed?
Your turn1 of 2
Find E[total claims] when the number of claims N is Poisson with mean 36 and each claim averages 1333 dollars.