University:
University of Houston
Course:
MATH 1313 | Linear Algebra
Academic year:

2022
Views:

138

Pages:

15
Author:

sabrinadiaz300q

Jointly Distributed Random Variables Suppose that a fair, 6-sided die is rolled. Let X indicate the event that an even number is rolled (i.e. X = 1 if an even is rolled and 0 if not). Let Y indicate the event that a 4, 5, or 6 is rolled (i.e. Y = 1 if the roll is 4, 5, or 6, and Y = 0 if not). What is the probability that X and Y are both 1 (i.e. P(X = 1,Y = 1) ) ? "and" P(4or * 6) = 2/6 = 1/3 What about P * (X = 0Y = 1) deg ? Y P(5) We can display all the probabilities associated, with these random variables in table (called a contingency table). (odd) (even) (1,2,3) 01/3 0 بالا پرا کا P( Y = 0 ,) (4,5,6) 14 Уз 2 1/2 marginal distr. Popper 07 3. For Popper #2, are the variables dependent or independent? a dependent b. independent Expected Values, Covariance, and Correlation Expectation: For joint random variables X and Y, the expected values are: μ_x = E[X] = ∑_(x,y)∈X x * f(x,y) = ∑_x x * f_x(x), and μ_y = E[Y] = ∑_(x,y)∈Y y * f(x,y) = ∑_y y * f_y(y) E[g(X,Y)] = ∑_(x,y)∈X g(x,y) * f(x,y) contingency table with values Example: Use the contingency table from the last example to calculate E[X], E[Y], and E[XY]. E[X] = 1 * 3/30 + 2 * 7/30 + 3 * 8/30 + 4 * 12/30 = 89/30 E[Y] = 1 * 10/30 + 2 * 20/30 = 50/30 = 5/3 E[XY] = 1 * 1 * 1/30 + 1 * 2 * 2/30 + 2 * 1 * 2/30 + 2 * 2 * 5/30 + 3 * 1 * 4/30 + 3 * 2 * 4/30 + 4 * 1 * 3/30 + 4 * 2 * 9/30 = 149/30 Given two random variables X and Y, the covariance of X and Y is given by cov(X,Y) = E[XY] - E[X]E[Y] = 149/30 - (89/30)(5/3) = 45/90 and the correlation coefficient of X and Y is given by ρ = cov(X,Y) / (σ_x * σ_y) Compute the correlation coefficient in the previous example. σ_x = √Var(X) = √E[X²] - E[X]² = √(1.0322) = 1.016 σ_y = √Var(Y) = √E[Y²] - E[Y]² = √(.2222) = .4714 corr = ρ = r = (45/90) / (1.016 * .4714) = .0464