University:
York University
Course:
MATH 1131 | Introduction to Statistics I
Academic year:

2022
Views:

255

Pages:

5
Author:

Niklas

Let X and Y be a pair of jointly distributed discrete random variables with values {x1, ...} and {y1, ...}. The conditional probability distribution of the random variable Y, given that the random variable X takes the value x, is: P(Y=y|X=x) = P(X=x, Y=y) / P(X=x) It expresses the probability that Y takes the value y, as a function of y, when the value x is fixed for X. Example: The table that contains all the joint probabilities is the joint probability function of the pair (M, S). From the joint probability table, we compute the conditional probability of the random variable M conditioning on the values S=0, S=1, S=2, and S=3. So the number of conditional distributions we can obtain conditioning on a single value for one of the two rvs in this case is 8. Joint expected value: Let X and Y be two rvs assuming values {x1, ...} and {y1, ...} with joint probability function p(x, y). We define the joint expected value as: E(XY) = ΣΣ x_i * y_j * p(x_i, y_j) Section 1 Definition: Cov(X,Y) = E[(X - E(X))(Y - E(Y))] Calculation: Cov(X,Y) = ΣΣ (xi - μx)(yi - μy) * p(xi, yi) Properties of covariance: The covariance can be obtained by multiplying all the terms in the function k and applying the expected values. Cov(X,Y) = E(XY) - E(X)E(Y) Cov(aX+b, cY+d) = ac * Cov(X,Y) Section 2 Example: Tossing a coin Table: M (1, 2, 3, 4) S (0, 1, 2, 3, 4, 5, 6, 7, 8) Calculations: E(M) = Σi Mi * pi = 2.5 E(S) = Σj Sj * pj = 5.5 Cov(M,S) = E(MS) - E(M)E(S) = 2 - 2.5 * 5.5 = -13.75 Section 1 Heading: OUTCOMES Table: A table listing all possible outcomes of rolling two dice (1-1, 1-2, ..., 6-6). Section 2 Calculations: P(H ≤ 2) = P(H=1) + P(H=2) = 4/96 + 3/96 = 1/24 P(S|H=4) = P(S=1|H=4) + P(S=2|H=4) + P(S=3|H=4) + P(S=4|H=4) = 0/16 + 2/16 + 3/16 + 4/16 = 9/16 P(H|S=6) = P(H=1|S=6) + P(H=2|S=6) + P(H=3|S=6) + P(H=4|S=6) = 1/16 + 2/16 + 3/16 + 4/16 = 10/16 E(H) = Σi Mi * Pi = 4*(4/16) + 2*(3/16) + 3*(5/16) + 4*(4/16) = 50/16 = 25/8 E(S) = Σj Sj * Pj = 1*(4/16) + 2*(3/16) + 3*(5/16) + 4*(4/16) + 5*(2/16) + 6*(3/16) + 7*(2/16) + 8*(4/16) = 435/8