Econometrics: Marginal probability
Econometrics: Marginal probability ECO 5100
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This 2 page Class Notes was uploaded by Rahul Bose on Wednesday June 22, 2016. The Class Notes belongs to ECO 5100 at Wayne State University taught by Arjun in Summer 2016. Since its upload, it has received 15 views. For similar materials see Econometrics in Economcs at Wayne State University.
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Date Created: 06/22/16
Chapter 2 : Probablity The probability density function (pdf) of X summarizes the information concerning the possible outcomes of X and the corresponding probabilities: f(xj ) = pj , j = 1, 2, ..., k, with f(x) = 0 for any x not equal to xj for some j. f(x) is the probability that X takes on the particular value x. When dealing with more than one random variable, we use fX to denote the pdf of X, fY to denote the pdf of Y ,etc. It is simple to compute the probability of any event involving the random variable with the pdf. e.g. suppose X is the number of free throws made by a basketball player out of two attempts, so that X can take on the three values 0,1,2. Assume that the pdf of X is given by f(0) = .20, f(1) = .44, and f(2) = . 36. Then we can use it to calculate the probability that the player makes at least one free throw: P(X ≥ 1) = P(X = 1) + P(X = 2) = .44 + .36 = .80 A variable X is a continuous random variable if it takes on any real value with zero probability. We use the pdf of a continuous random variable only to compute events involving a range of values. e.g. if a and b are constants where a < b, the probability that X lies between the numbers a and b, P(a ≤ X ≤ b) is the area under the pdf between points a and b, as shown in Figure B.2. This is the integral of the function f between a and b. The entire area under the pdf must always equal one. For a random variable X, its cumulative distribution function (cdf) is defined for any real number x by F(x) = P(X ≤ x). (1) For discrete random variable, (1) is obtained by summing the pdf over all values xj such that xj ≤ x. For a continuous random variable, (1) is the area under the pdf, to the left of the point x. Because F(x) is simply a probability, it is always between 0 and 1. A cdf is an increasing (or at least nondecreasing ) function of x since if x1 < x2, then P(X ≤ x1) ≤ P(X ≤ x2), i.e. F(x1) ≤ F(x2). For a continuous random variable X, for any number c, P(X ≥ c) = P(X > c) = 1 − F(c) And for any numbers a < b, P(a < X < b) = P(a ≤ X ≤ b) = P(a ≤ X < b) = P(a < X ≤ b) = F(b) − F(a) Let X and Y be discrete random variables. Then (X, Y ) have a joint distribution, which is fully described by the joint probability density function of (X, Y ): fX,Y (x, y) = P(X = x, Y = y), where the right hand side is the probability that X = x and Y = y. The joint pdf for a continuous random vector (X, Y ), f(x, y), is defined as P(a ≤ X ≤ b, c ≤ Y ≤ d) = Z d c Z b a f(x, y)dxdy Let X denote the number of houses that agent 1 will sell in a month and let Y denote the number of houses agent 2 will sell in a month. An analysis of their past monthly performance has the following joint probabilities: Table: Bivariate Probability Distribution X 0 1 2 Y 0 .12 .42 .06 1 .21 .06 .03 2 .07 .02 .01 Then, e.g. the probability that agent 1 sells 0 houses and agent 2 sells 1 house in the month is P(0, 1) = .21. And we can calculate the marginal probabilities by summing across row or down columns. e.g. P(X = 0) = P(0, 0) + P(0, 1) + P(0, 2) = .12 + .21 + .07 = .4 The marginal probability distribution of X is x P(x) 0 .4 1 .5 2 .1 Similarly, P(Y = 0) = P(0, 0) + P(1, 0) + P(2, 0) = .12 + .42 + .06 = .6 The marginal probability distribution of Y is y P(y) 0 .6 1 .3 2 .1
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