Tables that hold two traits at once

Computing · Machine Learning · ages 18-19

Name ______________________ Date ____________

A two-way table shows 40 students by gender and sport. Women: 8 soccer, 8 basketball, 4 lacrosse. Men: 4 soccer, 12 basketball, 4 lacrosse. What is the marginal proportion of all students who play basketball?
- 0.20
- 0.30
- 0.50
- 0.80
In the 40-student table, what is the marginal share of basketball players?
- 0.20
- 0.50
- 0.80
In general, P(A given B) and P(B given A) are the same number.
Circle one: True False
In the 40-student table, how many students play lacrosse?
Answer: ______________
True or false: two discrete random variables are independent exactly when every joint entry in the table equals the product of the corresponding two marginal probabilities.
Circle one: True False
A joint table gives P(X=1,Y=0)=0.3, P(X=1,Y=1)=0.1, P(X=2,Y=0)=0.2, P(X=2,Y=1)=0.4. Compute the expected value E of X.
Answer: ______________
Two random variables X and Y have joint probabilities P(X=0,Y=1)=0.2, P(X=0,Y=2)=0.1, P(X=1,Y=1)=0.3, P(X=1,Y=2)=0.4. Compute P(X=1 given Y=2). Round to two decimals if needed.
Answer: ______________
Using the joint table P(X=1,Y=0)=0.3, P(X=1,Y=1)=0.1, P(X=2,Y=0)=0.2, P(X=2,Y=1)=0.4, compute the expected value E of X plus Y.
Answer: ______________
A table gives P(Red, Circle)=0.12 with P(Red)=0.40 and P(Circle)=0.30, and every other cell matches its product. Are color and shape independent?
- No, because one cell looks small
- Yes, because every cell equals its row times column product
- Cannot tell without the grand total
A joint density is f(x,y)=c(x+y) on the unit square, where 0 is less than x and x is less than 1 and 0 is less than y and y is less than 1. The constant c makes the total probability equal 1. Compute P(X less than 0.5). Round to three decimals if needed.
Answer: ______________

Answer key

For grown-ups. Fold this page away before handing over the rest.

Tables that hold two traits at once W1-mt__Rj-hOSMEw-s1

0.50 · The marginal distribution ignores gender, so add the basketball column and divide by the grand total: 20 out of 40, which is 0.50.
0.50 · Add the basketball column and divide by everyone: 20 out of 40 is 0.50.
False · Both divide the same overlap by different bases, so they usually differ.
8 · Add the lacrosse column: 4 women plus 4 men makes 8.
True · That product rule is the definition of independence for discrete variables. If even one cell breaks the rule, the variables are dependent.
1.6 · First sum each row of X to get its marginal: P(X=1)=0.4 and P(X=2)=0.6. Then E of X is 1 times 0.4 plus 2 times 0.6, which is 1.6.
0.8 · Restrict to the column where Y=2, then divide the joint entry by that column total: 0.4 / 0.5 = 0.8.
2.1 · E of X is 1.6 and E of Y is 0.5, and the expectation of a sum is the sum of the expectations even when the variables are dependent. So E of X plus Y is 2.1.
Yes, because every cell equals its row times column product · 0.40 times 0.30 is 0.12, and with every cell matching, the traits are independent.
0.375 · The double integral of (x+y) over the unit square is 1, so c=1. The marginal density of X is x plus 1/2, and integrating that from 0 to 0.5 gives 3/8, which is 0.375.