← Stat 110: Probability

Unit 8: Transformations and the Beta-Gamma Family

Turning one random variable into another, sliding one density across another to add them, and the Beta-Gamma pair behind the Poisson process.

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.

8.1 Transformations of a Random Variable

  1. How it worksHow does the Jacobian correction factor account for stretching and squeezing when transforming a PDF?
  2. Common mistakeWhy isn't the PDF of g(X) just the PDF of X plugged into g?
  3. Worked exampleFind the PDF of Y equal to X2 when X is uniform on the interval from 0 to 1.
  4. Going furtherWhat happens to a transformation's Jacobian when the function g is not monotonic?
Your turn1 of 2
X is uniform on 0 to 1. Let Y = X2. Write the PDF of Y, fY(y), as a function of y.

8.1 The Log-Normal Distribution

  1. How it worksHow does exponentiating a Normal random variable produce the Log-Normal's skewed shape?
  2. Worked exampleFind E[Y] for Y equal to eX when X is Normal with mean 0 and variance 1.
  3. Going furtherWhat happens to the Log-Normal distribution's skew as the underlying Normal's variance grows?
Your turn1 of 2
Find E[Y] for Y = eX when X is Normal with mean 1 and variance 3.42.

8.2 The Probabilistic Method

  1. How it worksHow does an expected count below or above a target guarantee some outcome reaches that target?
  2. Worked exampleFind why an 8-player round robin tournament must include a player with at least 4 wins, using the average win count.
  3. Going furtherWhat happens to the probabilistic method's guarantee when the average of a discrete quantity is already a whole number?
Your turn1 of 2
Find the average number of wins per player in a round-robin tournament of 22 players (each pair plays exactly once).

8.2 Convolutions: Adding Two Random Variables

  1. How it worksHow does the convolution integral sum over every way two values could combine to a given total?
  2. Worked exampleFind the PDF of Z equal to X plus Y at z=1 when X and Y are both independent Uniform(0,1) random variables.
  3. Common mistakeWhy doesn't the sum of two independent Uniform random variables stay uniformly distributed?
  4. Ties togetherHow does the convolution of two independent Poissons connect back to their sum being Poisson itself?
Your turn1 of 2
Find the PDF of Z = X + Y at z = 2.77 when X and Y are both independent Uniform(0, 15) random variables (with z less than b).

8.3 The Beta Distribution

  1. How it worksHow do the two shape parameters a and b let the Beta distribution take on wildly different shapes?
  2. Common mistakeWhy isn't a Beta(2,2) distribution the same shape as a Beta(5,5) even though both are symmetric?
  3. Worked exampleFind the mean of a Beta distribution with parameters a=2 and b=3.
  4. Going furtherWhat happens to the Beta distribution's shape as both parameters a and b grow very large together?
Your turn1 of 2
Find the mean of a Beta distribution with parameters a = 7 and b = 7.

8.3 Bayes' Billiards and the Beta-Binomial Connection

  1. How it worksHow does conditioning on the position of one uniform ball among n others connect to a Beta-Binomial sum?
  2. Worked exampleFind P(exactly 3 of 6 other uniform points land left of a 7th uniform point) using the Beta-Binomial identity.
  3. Ties togetherHow does Bayes' billiards connect the Beta integral to the discrete uniform distribution over ranks?
Your turn1 of 2
Find P(exactly k of 15 other uniform points land left of the (n+1)th point) using the Beta-Binomial identity — this probability is the same for every value of k from 0 to n.

8.4 The Gamma Distribution

  1. How it worksHow does summing several independent exponential waiting times build the Gamma distribution?
  2. Worked exampleFind the expected total waiting time for the 5th event in a Poisson process with rate 2 events per minute.
  3. Common mistakeWhy doesn't a Gamma distribution with a whole-number shape parameter reduce to a single Exponential?
  4. Going furtherWhat happens to the Gamma distribution's shape as its shape parameter grows very large?
Your turn1 of 2
Find the expected total waiting time for the 11th event in a Poisson process with rate 3.42 events per minute.

8.4 Poisson Processes

  1. How it worksHow do independent exponential gaps between events assemble into a full Poisson process over time?
  2. Worked exampleFind the probability of at least 2 raindrops hitting a patch in 10 seconds if the rate is 0.3 drops per second.
  3. Common mistakeWhy doesn't a Poisson process require events to arrive at evenly spaced intervals?
  4. Ties togetherHow does a Poisson process connect the Poisson distribution's event count to the Gamma distribution's wait?
Your turn1 of 2
Find the probability of at least 2 raindrops hitting a patch in 15 seconds if the rate is 0.379 drops per second.

8.5 The Beta-Gamma Relationship (Bank-Post Office)

  1. How it worksHow does the bank-post office story turn a Gamma split into independent Beta and Gamma random variables?
  2. Worked exampleFind the distribution of X divided by X plus Y when X is Gamma(2,lambda) and Y is Gamma(3,lambda) independently.
  3. Going furtherWhat happens to the bank-post office identity when the two independent Gammas share the same shape parameter?
Your turn1 of 2
X is Gamma(7, lambda) and Y is Gamma(7, lambda), independent. By the bank-post office story, X/(X+Y) is Beta(a,b). Find its mean.