Random Variables (Chapter 2) Random variable = A real-valued function of an outcome X = f(outcome) Domain of X: Sample space of the experiment. Ex: Consider an experiment consisting of 3 Bernoulli trials. Bernoulli trial = Only two possible outcomes – success (S) or failure (F). • “IF” statement: if … then “S” else “F” • Examine each component. S = “acceptable”, F = “defective”. • Transmit binary digits through a communication channel. S = “digit received correctly”, F = “digit received incorrectly”. Suppose the trials are independent and each trial has a probability ½ of success. X = # successes observed in the experiment. Possible values: Outcome Value of X (SSS) (SSF) (SFS) … … (FFF) Random variable: • Assigns a real number to each outcome in S. • Denoted by X, Y, Z, etc., and its values by x, y, z, etc. • Its value depends on chance. • Its value becomes available once the experiment is completed and the outcome is known. • Probabilities of its values are determined by the probabilities of the outcomes in the sample space. Probability distribution of X = A table, formula or a graph that summarizes how the total probability of one is distributed over all the possible values of X. In the Bernoulli trials example, what is the distribution of X? 1 Two types of random variables: Discrete rv = Takes finite or countable number of values • Number of jobs in a queue • Number of errors • Number of successes, etc. Continuous rv = Takes all values in an interval – i.e., it has uncountable number of values. • Execution time • Waiting time • Miles per gallon • Distance traveled, etc. Discrete random variables X = A discrete rv. • Probability mass function (PMF) of X = Probability distribution of X. • Notation: p(x) = P(X = x) = probability that the rv X takes the value x. • Once we have the PMF, we can compute any probability of interest. • 0 ≤ p(x) ≤ 1, total probability = ∑x p(x) = 1 Ex: “Pick Six” in Texas Lottery. In this Lottery, a player picks six numbers from the numbers 1 through 50 with no repetitions and pays $1.00. On Wednesday evenings, the Texas State Lottery Commission televises one of their employees randomly picking six balls without replacement, each with a number from 1 to 50 on it, from a large hopper. The player is paid if his/her number matches with the selected balls for three or more numbers. Suppose X = # of matches a player has with the selected balls. (a) What is the PMF of X? Value of X # of outcomes in sample space p(x) (# of matches) that give this value of X 0 = 0.44422536452 1 = 0.41005418264 2 = 0.12814193207 3 = 0.01666886921 4 = 0.00089297514 5 = 0.00001661349 6 = 0.00000006293 Total 1.00000000000 2 In this example, in general, p(x) = P(X = x) = (b) What is the probability of winning nothing? Cumulative distribution function (CDF) of X is defined as F(x) = P(X ≤ x) = ∑ y≤x p(y) • Gives the total probability on the left of x or the probability that the observed value of X is at most x. • F(x) is non-decreasing. • Jumps by p(x) at the point x. • F(– ∞) = 0, and F(+ ∞) = 1 • P( a < X ≤ b) = F(b) – F(a) Ex: Consider a rv X whose CDF is the following. 0 x < 1 1/6 1≤ x < 2 2/6 2≤ x < 3 F(x) = 3/6 3≤ x < 4 4/6 4≤ x < 5 5/ 6 5 ≤ x < 6 16≤ x (a) What is the PMF of X? (b) Find P(X will be greater than 3, but no more than 5). 3 Random vectors and joint distributions If X, Y = random variables then (X, Y) = random vector. It has joint PMF p(x, y) = P{(X, Y) = (x, y)} = P(X = x and Y = y) • 0 ≤ p(x, y) ≤ 1 • ∑(x,y) p(x, y) = total probability = 1 • P[(X, Y) ∈ A] = ∑(x,y)∈A p(x, y) From the joint PMF, we can obtain the marginal PMF’s of X and Y: From the marginal PMF’s, we may not be able to obtain the joint PMF. Ex: Consider a program with two modules, having module execution times X and Y minutes, respectively. The following table describes their joint PMF: p(x,y) y=1 y=2 y=3 y=4 x=1 1/4 1/161/16 1/8 x=2 1/16 1/8 1/4 1/16 (a) Find the probability that P(Y ≥ X). (b) Suppose that the two modules are run concurrently. Find the probability that the total execution time of the program is 3 minutes. 4 (c) Find the marginal PMF of X. Independent random variables Random variables X and Y are independent if for every pair (x, y) p(x, y) = i.e., {X = x} and {Y = y} are independent events for all (x, y). • Two rv’s are independent if their joint PMF is the product of their marginals. • To show independence, verify the above equality for all (x, y). • To show dependence, find one pair (x, y) that violates it. Ex cont’d: The marginal PMF’s of X and Y are: x 1 2 y 1 2 3 4 P(X=x) 0.5 0.5 P(Y=y) 5/16 3/16 5/16 3/16 Are X and Y independent? Note: The concept of joint distribution and independence can be extended to the case of more than two rv’s. See sections 2.8-2.9 of the text. 5.