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Quantitative Methods II 360-255-LW Vincent Carrier Discrete Random Variables A random variable X is a variable whose value depends on the outcome of a random experiment. Let X be a random variable taking values in some set DX . If DX only contains natural numbers, then we say that X is a discrete random variable. Example: The following random variables X are discrete random variables. a) X : number obtained by throwing a die DX = {1, 2, 3, 4, 5, 6} b) X : number of throws of a coin to get a head DX = {1, 2, 3, 4, . . .} c) X : number of plane crashes in a year DX = {0, 1, 2, 3, 4, . . .} It is customary to distinguish between the random variable X with a capital letter and its observed value x with a small letter. To each random variable X is associated a probability distribution p : DX → [0, 1] such that p(x) = P (X = x) for x ∈ DX . A probability distribution always satisfies the conditions 1) 0 ≤ p(x) ≤ 1 for x ∈ DX and 2) X p(x) = 1. x∈DX A probability distribution can be expressed in a table, or with the help of a formula, as the examples below illustrate. Example: Let X : number of children in a random family of a given village. Assume that it has the probability distribution x 0 1 2 3 4 Total p(x) 0.1 0.3 0.4 0.15 0.05 1 A probability distribution can be illustrated by a histogram: p(x) 6 0.4 0.3 0.2 0.1 0 1 2 3 x 4 Example: Consider an urn with 9 green balls and 7 yellow balls. Assume that 4 balls are picked without replacement. Let X : number of green balls picked. Then 9 7 x 4−x p(x) = 16 4 for x = 0, 1, 2, 3, 4. Example: Same as above, except with replacement. Then x 4−x 4 9 7 p(x) = x 16 16 for x = 0, 1, 2, 3, 4. The probability distributions of the last two examples are illustrated below. p(x) p(x) 6 6 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 1 2 3 Without Replacement 4 x 0 1 2 3 With Replacement 4 x