Report

Chapter 5.1 & 5.2: Random Variables and Probability Mass Functions Chris Morgan, MATH G160 [email protected] February 1, 2012 Lecture 10 1 2 Random Variables -A random variable (RV) is a real valued function whose domain is a sample space. - We will usually denote random variables by X, Y, Z and their respective values by x, y, z 3 Random Variables Discrete Random Variables Continuous Random Variables •Random variables that take on a finite (or countable) number of values. •Random variables that take on values in a continuum or infinitely many values. –Sum of two dice (2,3,4,…,12) –Number of children (0,1,2,…) –Number in attendance at the movies –Number of hired employees - Number of students coming to class –Height –Weight –Time - Time you can hold your breath - Lifetime of your cell phone batter 4 Random Variables Toss a fair coin 4 times. Suppose we are interested in the random variable X = number of heads. Outcome T T T T Combination 4C0 X 0 P(x) 1/16 1 4/16 T T T H 4C1 T H H T 4C2 2 6/16 T H H H 4C3 3 4/16 H H H H 4C4 4 1/16 5 Probability Mass Function (PMF) The values on the right of the table above are called the Probability Mass Function of the random variable X : p( x) P( X x) The probability of x = the probability(X = one specific x) A probability mass function (p.m.f) is a function which describes the probabilities a discrete type random variable will take on for any given value. They can be used to calculate the probabilities corresponding to an event relating to a random variable. 6 Probability Mass Function (PMF) We can take the PMF, graph it, and display it in the form of a histogram. 7 6 5 4 3 2 1 0 1 2 3 4 5 7 Probability Mass Function (PMF) x f(x) 2 1/36 3 2/36 4 3/36 5 4/36 6 5/36 7 6/36 8 5/36 9 4/36 10 3/36 11 2/36 12 1/36 P(X < 4) = P(3 < x < 8) = P(3 ≤ x ≤ 8) = 8 Probability Mass Function (PMF) x f(x) 2 1/36 3 2/36 4 3/36 5 4/36 6 5/36 7 6/36 8 5/36 9 4/36 10 3/36 11 2/36 12 1/36 P(x < 8 | x < 10) = P(x > 3) = P(4 < x < 7 | x < 9) = 9 Basic Properties of a PMF 1. p( x) 0x R (PMFs are nonnegative) 2. There are only finitely (or countably infinitely many) x’s for which: p( x) 0 3. p(x) 1 10 Fundamental Probability Formula How do you compute probabilities for a random variable X? Welll….. We have the PMF that tell us the probability that the random variable X takes on the specific value x. Sometimes, you may be interested in a whole range of possible values of X. For instance, the Pr(1 ≤ x≤ 3) 11 Example I toss a fair coin 4 times. What the probability of getting at most two most? P(x ≤ 2) = ? 1 4 6 11 16 16 16 16 P(x ≥ 1) = 1 – P(x=0) = 1 - 1/16 = 15/16 X 0 P(x) 1/16 1 4/16 2 6/16 3 4/16 4 1/16 12 Example I can also write this PMF as a function rather than a chart: 1 16 4 16 6 f p ( x) 16 4 16 1 16 0 For x = 0 For x = 1 For x = 2 X 0 P(x) 1/16 1 4/16 2 6/16 3 4/16 4 1/16 For x = 3 For x = otherwise 13 Fundamental Probability Formula Suppose X is a discrete RV and that A is a set of real numbers. Then: P( X A) p( x) xA In words: the sum of the probability mass function over all possible value you are interested in E( X ) x * P( X x) xA 14 Practice Problem #1 A partially eaten bag of M&M’s contains 2 red, 5 blue, and 3 green M&M’s. You and your buddy decide to place a bet. You will choose two M&M’s at random without replacement. For every red M&M you win $5, for every green M&M you win $1 and you do not win anything for a blue M&M. Let X be the amount of money you will win. 15 Practice Problem #1 [2 red, 5 blue, and 3 green M&M’s….pick two] Let X be the amount of money you will win. Begin by writing down the PMF: Colors X p(x) R and R 10 1/45 R and G 6 2*3 = 6/45 R and B 5 2*5 = 10/46 B and G 1 5*3 = 15/45 B and B 0 5C2 = 10/45 G and G 2 3C2 = 3/45 16 Practice Problem #1 What is the probability you will win at least five dollars? P(X ≥ 5) = 1/45 + 6/45 + 10/45 = 17/45 Colors X p(x) R and R 10 1/45 R and G 6 2*3 = 6/45 R and B 5 2*5 = 10/46 G and G 2 3C2 = 3/45 B and G 1 5*3 = 15/45 B and B 0 5C2 = 10/45 17 Practice Problem #1 If you win something, what is the probability it will be worth at least five dollars?? P(X ≥ 5 | X ≠ 0) = 1 6 10 17 P( X 5andx 0) 45 45 45 45 17 10 35 35 P( x 0) 1 45 45 18 Practice Problem #1b X p(x) 10 1/45 6 6/45 5 10/46 1 15/45 0 10/45 2 3/45 E(X)= E(X)2= E(X2)= 19 Practice Problem #2a In a simple game, two fair coins are tossed and the payoff is to be determined from the outcome. The payoff strategy is as follows: Win $5 for each head, lose $10 for two tails. Let X denote your winnings if you play this game once. 20 Practice Problem #2a Win $5 for each head, lose $10 for two tails. Write out the PMF for x: 2 heads 1 tail, 1 head 2 tails X 10 5 -10 p(x) 1/4 1/2 ¼ E(X) = 10(1/4) + 5(1/2) – 10(1/4) = 2.5 21 Practice Problem #2b In a simple game, two fair coins are tossed and the payoff is to be determined from the outcome. The payoff strategy is as follows: Lose $5 for two tails Win $5 for two different Lose $10 for two heads Let X denote your winnings if you play this game once. 22 Practice Problem #2b Calculate the PMF for this payoff strategy: Calculate the expected payoff E(X) = Which payoff would you rather play with, a or b, and why?? 23