Econ 3120 1st Edition Lecture 1Outline of Last Lecture I. Syllabus reviewOutline of Current Lecture I. Random VariablesII. Continuous DistributionsIII. Multivariate DistributionsCurrent Lecture1 Random Variables: A random variable is a variable that “takes on numerical values and has an outcomethat 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 thatthe 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= These notes represent a detailed interpretation of the professor’s lecture. GradeBuddy is best used as a supplement to your own notes, not as a substitute.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 = 12The cumulative distribution function for a continuous random variable whose pdf is f(t) is given by F(x) = P(X ≤ x) = ˆ x −∞ f(t)dt Properties of F(x) for a continuous random variable: 1. F(−∞) = 0,F(∞) = 1 2. F(x) is nondecreasing in x, i.e., if a < b, then F(a) ≤ F(b) Other properties of the cdf: 4. P(a ≤ X ≤ b) = F(b)−F(a) 5. P(X > a) = 1−P(X ≤ a) = 1−F(a) 6. f(x) = dF(x) dx where the derivative of F(x) exists.Multivariate Distributions If we have two random variables, X and Y, we write the probability that X and Ytake on particular values x and y as P(X = x,Y = y).The joint probability distribution over X and Y is given by f(x, y). For discrete random variables, f(x, y) represents P(X = x,Y = y) For continuous random variables, we have to define the probabilities over a range of X and Y (similar to the univariate case): P(a ≤ X ≤ b, c ≤ Y ≤ d) = ˆ b a ˆ d c f(x, y)dy dx Properties of the joint distribution for random variables: 1. f(x, y) ≥ 0 for x, y 2. ∑x ∑y f(x, y) = 1 (discrete) ´ x ´ y f(x, y)dy dx = 1 If X and Y are discrete random variables, the joint cumulative distribution of X and Y is given by F(x, y) = P(X ≤ x,Y ≤ Y) = ∑ s≤x ∑ t≤y f(s,t) for −∞ ≤ x ≤ ∞ −∞ ≤ y ≤ ∞ For a continuous random variable, the joint cumulative distribution of X and Y is given by F(x, y) = P(X ≤ x,Y ≤ Y) = ˆ y −∞ ˆ x −∞ f(s,t)ds dt for−∞ ≤ x ≤ ∞ −∞ ≤ y ≤ ∞ For continuous distributions, we also have the following relationship between the joint cumulativeand the joint probability distributions (density function): f(x, y) = ∂ 2 ∂ x∂ y F(x,