Random Variables(2 pages)
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- Lecture number:
- Lecture Note
- Cornell University
- Econ 3120 - Applied Econometrics
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Lecture 1 Outline of Last Lecture I. Syllabus review Outline of Current Lecture I. Random Variables II. Continuous Distributions III. Multivariate Distributions Current Lecture 1 Random Variables: A random variable is a variable that “takes on numerical values and has an outcome that is determined by an experiment.” A discrete random variable takes on “countably many” values. A continuous random variable takes on values that lie on a continuum. (e.g., an interval from 0 to 1). We denote the random variable by X (or some other capital letter), and the values the variable can take as x (or some other lowercase letter) The probability that a random variable X takes on a particular value x is denoted by P(X = x) I.1 Discrete Distributions A random variable X has a discrete distribution (or X is a discrete random variable) if X can take only a finite number of values x1, x2,..., xkor at most, a countably infinite sequence of values. The probability distribution function of a discrete random variable X is defined as the function f(x) = P(X = x)and represents the probability that the random variable X takes on each potential value of x. This is sometimes called the probability mass function (but only for a discrete random variable). Properties of f(x) for a discrete random variable: 1. 0 ≤ f(x) ≤ 1 for each value within the range of X 12. ∑x f(x) = 1, where the summation is over all values within the range of X The cumulative distribution function of a discrete random variable X is given by F(x) = P(X ≤ x) = ∑ t≤x f(t), −∞ < x < ∞ where f(t) is the probability distribution of X at point t. Properties of F(x): 1. F(−∞) = 0, F(∞) = 1 2. F(x) is nondecreasing in x, i.e., if a < b, then F(a) ≤ F(b) 1.2 Continuous Distributions A random variable X has a continuous distribution (or X is a continuous random variable) if X can assume any value in an interval General rule: With a continuous random variable, we assign zero probability to single points. In other words, P(X = x)= 0. The probability density function for a continuous random variable X is defined as f(x) such that P(a ≤ X ≤ b) = ˆ b a f(x)dx Unlike the case with discrete random variables, the values of f(x) have no useful meaning (note that f(x) 6= P(X = x) = 0). Instead, we assign probability to the integral of f(x) over a certain interval. Properties of f(x) for a continuous random variable: 1. f(x) ≥ 0 for −∞ < x < ∞ 2. ´ ∞ −∞ f(x)dx = 1 2 Econ 3120 1st Edition
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