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* A random variable is a rule that assigns one value to each point in a sample space for an experiment. * A random variable can be classified as discrete or continuous depending on the numerical values it assumes. 1. A discrete random variable may assume either finite or infinite sequence of values. 2. A continuous random variable may assume any numerical in an interval or collection of interval. 1. number of children in a family 2. Friday night attendance at a cinema 3. number of patients in a doctor's surgery 4. number of defective light bulbs in a box of ten. 1. 2. 3. 4. height of students in class weight of students in class time it takes to get to school distance traveled between classes Consider an experiment of tossing a coin three times. S = {HHH, HHT, HTT, HTH, THT, THH, TTH, TTT} Let X assign to each sample point on S the total number of head occurs. Then X is random variable with range space Rx = { 0, 1, 2, 3}, since range space is finite, X, is a discrete random variable. Couple plans to have 2 children. The random circumstance includes the 2 births, specifically the sexes of the 2 children. Let X assign to each sample point on S the number of girls. BOY BOY GIRL union BOY GIRL GIRL S = {BB, BG, GB, GG} X is a random variable with range Rx = {0, 1, 2}, since Rx is finite X is a discrete random variable. If X is a random variable, the function given by f(x) = P [ X = x ] for each X within the range of X is called the probability mass function (pmf ) of X. A function can serve as probability mass function of a discrete random variable X if and only if its values f(x), satisfies the following conditions: 1. f(x) 0 for all values of x; 2. of all f(x) is equal to 1. To express the probability mass function, we will construct a table that exhibits the correspondence between the values of random variables and the associated probabilities. Consider example # 1. The experiment consisting of three tosses of a coin, assume that all 8 outcomes are equally likely then the probability mass function for the total number of heads is: X 0 f(X) 1/8 1 3/8 2 3/8 3 1/8 To show that is a probability mass function (pmf): Condition 1: Notice that all f(x) are all greater than or equal to zero ; Condition 2: The sum of all f(x) is 1, that is: 1/8 + 3/8 + 3/8 + 1/8 = 1. Referring to example # 2. The experiment on the plan of the couple who wanted to have two children. There are 6 possible outcomes belonging to the sample space S. Let X assign to each sample point on S the number of girls. X is a random variable defined by a function f(x) = P [ X =x ], thus it would be a (pmf) such that: x f(x) 0 ¼ 1 ½ 2 ¼ Condition 1 is satisfied: for all f(x) it is greater than or equal to zero; Condition 2 is satisfied: the sum of all f(x) is equal to 1, that is (f (x)) = ¼ + ½ + ¼ = 1.