← Stat 110: Probability

Unit 4: Expectation

The single number that summarizes a random variable's center of mass — built to stay linear no matter how tangled the dependence underneath it gets.

8 topics · 30 questions

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4.1 Defining Expected Value

  1. How it worksHow does expected value weight every possible outcome by how likely it is?
  2. Worked exampleFind the expected winnings of a game paying 10 dollars on heads and nothing on tails, costing 4 dollars to play.
  3. Common mistakeWhy isn't expected value the outcome you'd most expect to see on any single trial?
Your turn1 of 2
A game pays 36 dollars if a fair coin lands heads and nothing on tails, and costs 13 dollars to play. Find the expected net winnings.

4.2 Linearity of Expectation

  1. How it worksHow does linearity of expectation hold even when the random variables being added aren't independent?
  2. Worked exampleFind the expected number of heads in 10 flips of a coin that lands heads with probability 0.3.
  3. Common mistakeWhy doesn't linearity of expectation require the random variables to be independent of each other?
  4. Going furtherWhat happens to linearity of expectation when you multiply a random variable by a negative constant?
Your turn1 of 2
Find the expected number of heads in 71 flips of a coin that lands heads with probability 0.619.

4.2 The Coupon Collector Problem

  1. How it worksHow does breaking the wait into geometric stages between new coupon types set up linearity of expectation?
  2. Common mistakeWhy doesn't the expected time to collect all n coupons grow proportionally with n?
  3. Worked exampleFind the expected number of cereal boxes needed to collect all 10 distinct prizes, one random prize per box.
  4. Going furtherWhat happens to the expected coupon-collecting time as the number of distinct coupon types grows from 10 to 100?
Your turn1 of 2
In the coupon collector problem with 38 distinct coupon types, once 5 distinct types have already been collected, find the expected number of additional draws needed to get a new (not yet seen) type.

4.2 The St. Petersburg Paradox

  1. How it worksHow does doubling the payout every time a fair coin lands tails again build an expectation that diverges?
  2. Common mistakeWhy doesn't an infinite expected value mean a rational person should pay any finite amount to play?
  3. Worked exampleFind the expected payout of a game paying 2 to the k dollars when the first head appears on flip k.
  4. Going furtherWhat happens to the St. Petersburg paradox when the casino can only pay out up to a finite cap?
Your turn1 of 2
A casino plays the St. Petersburg game (payout 2k dollars if the first head appears on flip k) but caps the payout at 215 dollars once flip 15 is reached. Find the expected payout.

4.3 The Fundamental Bridge Between Events and Expectation

  1. How it worksHow does the expectation of an indicator random variable become exactly the probability of its event?
  2. Worked exampleFind P(a random card drawn is a spade) by computing E of the indicator for drawing a spade.
  3. Ties togetherHow does the fundamental bridge connect indicator random variables back to linearity of expectation?
Your turn1 of 2
A deck has 76 cards total, 13 of which are spades. Find P(a random card drawn is a spade) by computing E of the indicator for drawing a spade.

4.4 Symmetry Arguments for Expectation

  1. How it worksHow does swapping the roles of identical random variables pin down an expectation without a sum?
  2. Worked exampleFind the expected position of a specific card in a shuffled deck of 52 cards using symmetry.
  3. Going furtherWhat happens to a symmetry argument for expectation when the random variables aren't identically distributed?
Your turn1 of 2
Find the expected position of a specific card in a shuffled deck of 39 cards using symmetry.

4.5 Expectation of the Named Discrete Distributions

  1. How it worksHow does linearity of expectation reduce a negative binomial's mean to r copies of a geometric's mean?
  2. Worked exampleFind the expected number of rolls until the first six on a fair die.
  3. Worked exampleFind the expected number of defective items in a batch of 200 with a 3 percent defect rate.
  4. Common mistakeWhy doesn't a Poisson distribution's expectation need any calculation once you know its rate parameter?
Your turn1 of 2
Find the expected number of rolls until the first success, for an event with probability 0.633 on each independent roll.

4.6 Variance and Standard Deviation

  1. How it worksHow does Var(X) equal E[X2] minus (E[X])2 by expanding the squared definition of variance?
  2. Worked exampleFind the variance of a fair six-sided die roll.
  3. Worked exampleFind the standard deviation of a binomial distribution with n equal to 100 and p equal to 0.2.
  4. Common mistakeWhy doesn't the variance of a sum of random variables just add up when the variables are dependent?
  5. Going furtherWhat happens to Var(aX plus b) when a random variable is scaled by a and shifted by b?
Your turn1 of 2
Find the standard deviation of a binomial distribution with n equal to 215 and p equal to 0.424.