Unit 7: Joint Distributions
More than one random variable at once — joint, marginal, and conditional distributions, 2D LOTUS, covariance, and the Multinomial and Multivariate Normal.
10 topics · 35 questions
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7.1 Joint, Marginal, and Conditional Distributions
- How it worksHow does summing a joint PMF over one variable produce the other variable's marginal distribution?
- Worked exampleFind the marginal distribution of X from a joint PMF table with P(X=1,Y=1)=0.2, P(X=1,Y=2)=0.1, P(X=2,Y=1)=0.3, P(X=2,Y=2)=0.4.
- Common mistakeWhy isn't the joint distribution recoverable just from the two marginal distributions alone?
- Going furtherWhat happens to a joint distribution's conditional slices when the two variables are independent?
Your turn1 of 2
A joint PMF has P(X=1,Y=1) = 0.189, P(X=1,Y=2) = 0.18, P(X=2,Y=1) = 0.236, and P(X=2,Y=2) equal to whatever remains (so all four sum to 1). Find the marginal P(X=1).7.1 Independence of Random Variables
- How it worksHow does a joint PMF or PDF splitting into a product of the two marginals define independence?
- Worked exampleFind whether X and Y are independent given the joint PMF 0.2, 0.2, 0.3, 0.3 at (1,1), (1,2), (2,1), (2,2).
- Common mistakeWhy doesn't zero covariance between two random variables prove they're independent?
Your turn1 of 2
X and Y are independent, with P(X=1) = 0.618 and P(Y=1) = 0.589. Find P(X=1 and Y=1).7.2 The Chicken-Egg Problem: Poisson Thinning
- How it worksHow does conditioning on the total count and flipping independent coins for each item produce Poisson thinning?
- Worked exampleFind the distribution of eggs that hatch when the number laid is Poisson with rate 10 and each hatches independently with probability 0.6.
- Going furtherWhat happens to the joint distribution of hatched and unhatched eggs once the thinning probability is fixed?
Your turn1 of 2
The number of eggs laid is Poisson with rate 22, and each hatches independently with probability 0.654. By Poisson thinning, the number that hatch is Poisson with rate lam·p. Find the expected number of eggs that DON'T hatch.7.3 2D LOTUS
- How it worksHow does 2D LOTUS sum or integrate g(x,y) against the joint PMF or PDF directly?
- Worked exampleFind E[XY] using 2D LOTUS for the joint PMF with P(X=1,Y=2)=0.3, P(X=2,Y=1)=0.3, P(X=2,Y=2)=0.4.
- Common mistakeWhy doesn't 2D LOTUS require X and Y to be independent to compute E[g(X,Y)]?
Your turn1 of 2
Using 2D LOTUS, find E[XY] for the joint PMF P(X=1,Y=2) = 0.154, P(X=2,Y=1) = 0.147, P(X=2,Y=2) = 0.19 (and P(X=1,Y=1) is whatever remains).7.4 Covariance and Correlation
- How it worksHow does dividing covariance by both standard deviations rescale it into the unitless correlation coefficient?
- Worked exampleFind Cov(X,Y) when E[XY]=2.8, E[X]=1.7, and E[Y]=1.6.
- Worked exampleFind the correlation between X and Y when Cov(X,Y)=0.08, SD(X)=0.46, and SD(Y)=0.49.
- Common mistakeWhy doesn't a correlation of zero between two random variables guarantee that they're independent?
- Going furtherWhat happens to correlation when both random variables are scaled by positive constants?
Your turn1 of 2
Find the correlation between X and Y when Cov(X,Y) = 0.111, SD(X) = 0.66, and SD(Y) = 0.773.7.4 The Variance of a Sum
- How it worksHow does expanding Var(X+Y) as E[(X+Y)2] minus (E[X+Y])2 reveal the covariance term?
- Worked exampleFind Var(X+Y) when Var(X)=4, Var(Y)=9, and Cov(X,Y)=2.
- Common mistakeWhy doesn't Var(X+Y) equal Var(X) plus Var(Y) whenever X and Y are dependent?
- Ties togetherHow does the variance-of-a-sum formula explain why the hypergeometric's variance differs from the binomial's?
Your turn1 of 2
Find Var(X+Y) when Var(X) = 7, Var(Y) = 7, and Cov(X,Y) = 0.43.7.5 The Variance of the Hypergeometric Distribution
- How it worksHow does the finite population correction factor fall out of summing covariances between dependent draws?
- Worked exampleFind the variance of the number of red balls in 5 draws without replacement from an urn of 10 red and 15 blue balls.
- Going furtherWhat happens to the hypergeometric variance as the population size N grows toward infinity?
Your turn1 of 2
Find the variance of the number of red balls in 5 draws without replacement from an urn of 15 red and 23 blue balls.7.6 The Multinomial Distribution
- How it worksHow does the multinomial distribution generalize the binomial from two outcomes to several?
- Worked exampleFind the probability of rolling two 1s, two 2s, and one each of 3, 4, 5, and 6 in 8 rolls of a fair die.
- Going furtherWhat happens to the marginal distribution of one category when a multinomial collapses to success versus everything else?
Your turn1 of 2
An experiment has three possible outcomes with probabilities 0.239, 0.23, and whatever remains. In 3 independent trials, find the probability of getting each of the three outcomes exactly once.7.7 The Multivariate Normal Distribution
- How it worksHow does a covariance matrix stretch and tilt independent standard Normals into a correlated Multivariate Normal?
- Worked exampleFind the shape of the Multivariate Normal's contour ellipses when X and Y have equal variance and correlation 0.8.
- Common mistakeWhy doesn't having two Normal marginals automatically make a pair jointly Multivariate Normal?
- Ties togetherHow does the Multivariate Normal's covariance matrix connect back to the single-variable Normal's variance?
Your turn1 of 2
Two jointly Normal variables X and Y each have variance 7 and correlation 0.619. Find the variance along the ellipse's major axis (the direction where they move together), which equals v(1+rho).7.8 The Cauchy Distribution
- How it worksHow does dividing one standard Normal by another produce the Cauchy distribution's heavy tails?
- Worked exampleFind the PDF of the Cauchy distribution formed by X divided by Y for independent standard Normals X and Y.
- Common mistakeWhy doesn't the law of large numbers pull the average of many Cauchy random variables toward a fixed value?
Your turn1 of 2
Write the PDF of the standard Cauchy distribution (the ratio of two independent standard Normals), as a function of x.