Pairs of a random variable

Transcription

1 Handout 8 Pairs of a random variable "Always be a little improbable." Oscar Wilde in the previous tutorials we analyzed experiments in which an outcome is one number. we'll start to analyze experiments in which an outcome is a collection of numbers; each number is a sample value of a random variable; thus we analyzes experiments that produce two random variables, X and Y. the results presented here can be generalized for a system of n random variables X 1, X 2,... X n For a pair of discrete random variables X and Y : the joint cumulative distribution function of a pair of two discrete random variables X and Y is: F X,Y (x, y) = P (X x, Y y) it has similar properties with the CDF corresponding to a single random varaiable: i) 0 F X,Y (x, y) 1 ii) F X (x) = lim F X,Y (x, y) y iii) F Y (y) = lim F X,Y (x, y) x iv) lim F X,Y (x, y) = lim F X,Y (x, y) = 0 x x v) if x x 1 and y y 1, then F X,Y (x, y) F X,Y (x 1, y 1 ) vi) lim F X,Y (x, y) = 1 x,y the joint probability mass function of a pair of two discrete random variables X and Y is: P X,Y (x, y) = P (X = x, Y = y) for independent random variables X, Y one has: P X,Y (x, y) = P (X = x, Y = y) = P (X = x) P (Y = y) keep in mind that {X = x, Y = y} is an event in an experiment, thus the sample space is now: S X,Y = {(x, y) : P X,Y (x, y) > 0}

2 the probability of the event A is: P (A) = (x,y) A P X,Y (x, y) the marginal probability mass functions are: P X (x) = y S Y P X,Y (x, y), P Y (y) = For a pair of continuous random variables X, Y : x S X P X,Y (x, y) The joint probability density function of the continuous random variables X and Y is a function f X,Y (x, y) with the property: and: F X,Y (x, y) = x y f X,Y (u, v)dudv P (a X b, c Y d) = F X.Y (b, d) F X.Y (b, c) F X.Y (a, d)+f X.Y (a, c) for independent random variables: f X,Y (x, y) = f X (x) f Y (y) The probability of an event A is: P (A) = f X,Y (x, y)dxdy the marginal density functions are: f X (x) = A f X,Y (x, y)dy, f Y (y) = An Example f X,Y (x, y)dx Mark and Lisa are two real estate agents. Let X and Y be the respective numbers of houses sold by them in a month. Based on past sales, we estimated the following joint probabilities for X and Y :

3 Thus, for example P (0, 1) = 0.21, meaning that the joint probability for Mark and Lisa to sell 0 and 1 houses, respectively, is Other entries in the table are interpreted similarly. Note that the sum of all entries must equal to 1. The marginal probabilities are calculated by summing across rows and down columns: This gives us the probability mass functions for X and Y individually: Thus, for example, the marginal probability for Mark to sell 1 house is 0.5. We have: P (X = 0 and Y = 2) = 0.07, but P (X = 0) = 0.4, and P (Y independent: = 2) = 0.1, hence, X and Y are not P (X = 0 and Y = 2) P (X = 0) P (Y = 2) We could be interested in the probability for having two houses sold (by either Mark or Lisa) in a month. This can be computed by adding the probabilities for all combinations of (x, y) pairs that result in a sum of 2: P (X + Y = 2) = P (0, 2) + P (1, 1) + P (2, 0) = 0.19 Using this method, we can derive the probability mass function for the variable X + Y :

4 Proposed problems Problem 1. (Buon's needle problem) A oor has parallel lines on it at equal distances L from each other. A needle of length l is dropped at random onto the oor. Find the probability that the needle will intersect a line. Problem 2. (The problem of meeting revisited) Two people agree to meet between 21 : 00 P.M. and 23 : 00 P.M., with the understanding that each will wait no longer than 15 minutes for the other. What is the probability that they will meet? Problem 3. The joint probability function for the random variables X and Y is given in the table. (a) Find the marginal probability functions of X and Y. (b) Find P (1 X < 3, Y 1). (c) Determine whether X and Y are independent Problem 4. The joint density function of two continuous random variables X and Y is: { cxy, 0 < x < 4, 1 < y < 5 f(x, y) = 0, otherwise a) Find the value of c b) Find P (1 < X < 2, 2 < Y < 3) c) Find the marginal distribution functions of X and of Y

5 Problem 5. Let the random variable X be the portion of a ood insurance claim for ooding damage to the house and Y the portion of the claim for ooding damage to the rest of the property. The joint density function of X and Y is given by f(x, y) = 3 2x y for 0 < x, y < 1 and x + y < 1. What are the marginal densities of X and Y?