List of Online Homework: Homework 6: Random Variables and Discrete Variables Homework7: Expected Value and Standard Dev of a Variable Homework8: The Binomial Distribution List of Online Quizzes: Quiz7: Basic Probability Quiz 8: Expectation and sigma. Quiz 9: Binomial Introduction Quiz 10: Binomial Probability 1
RANDOM VARIABLES DEFINITION: A random variable is an uncertain numerical quantity whose value depends on the random outcome of an experiment. We can think of a random variable as a rule that assigns one (and only one) numerical value to each outcome of a random experiment. Think of the capital letter X as random, the value of the variable before it is observed. Think of x as known, a particular value of X that has been observed. DEFINITION: A discrete random variable can assume at most a finite or infinite but countable number of distinct values. A continuous random variable can assume any value in an interval or collection of intervals. Let's Do It! 1 a. Consider the experiment of two six-sided dice. Define the random variable x as the sum of the results. What are the values the random variable x will assume? b. Consider the experiment of tossing three coins. Define the random variable as the number of heads. What are the values the random variable x will assume? c. A sample of 9 cell phone users is selected. Subjects were asked if they are satisfied with their services. Define the random variable as the number of subjects who are satisfied with their services. What are the values the random variable x will assume? d. A teacher gives a 30 minutes tests. Define the random variable as the time it takes a student to finish the test. What are the values the random variable x will assume? e. Students final grade is the percentage of their course work (tests, homework, and quizzes) to the overall course work. Define the random variable as the final grade of a student. What are the values the random variable x will assume? 2
Discrete Random Variables Value of X X = x x1 x2... xk Probability P( X = x ) p1 p2... pk p if x = x p if x = x We can also represent this function as: p( x) p if x = x 1 1 2 2 k k DEFINITION: The probability distribution of a discrete random variable X is a table or rule that assigns a probability to each of the possible values of the random variable X. The values of a discrete probability distribution (the assigned probabilities) must be between 0 and 1 and must add up to 1, that is, p i 1. Let's Do It! 3 (2min) Determine which of the following represents a probability distribution. Explain: a. x 0 1 2 3 P(x) 0.26 0.32 0.42 0 b. p( x) 1, X 4, 2, 4 x c. p( x). 5 x, x 0. 75, 0, 0. 25 3
Example People in a Household Let X be the number of people in a household for a certain community. Value of X = x 1 2 3 4 5 6 7 Probability P[X = x].20.32.18.15.07.03? (a) What must be the probability of 7 people in a household for this to be a legitimate discrete distribution? (b) Display this probability distribution graphically. p ro b a b i l i ty p (x ).4.3.2 probability p b a b i l ibar ty hgraph i s to g ra m p (x ).4.3.2 1 2 3 4 5 6 7 x = # p e o p l e 1 2 3 4 5 6 7 x = # p e o p l e (c) What is the probability that a randomly chosen household contains more than 5 people? P X 5 (d) What is the probability that a randomly chosen household contains no more than 2 people? P X 2 (e) What is The probability that a randomly selected household has more than 2 but at most 4 people? P 2 X 4 4
Let's Do It! 4 Sum of Pips Craps game = rolling 2 fair dice. Let X be the sum of the values on the two dice. the possible pairs of faces of the 2 dice. S = { (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6) } (a) Give the probability distribution function of X, then present the probability distribution function graphically. Probability Value of X X=x Probability P(X=x) (b) Find the P( X > 7 ). X = sum (c) What is the probability of rolling a seven or an eleven on the next roll of the two dice? (d) What is the probability of rolling at least a three on the next roll of the two dice? (Use the complement rule.) 5
Expected Value of a discrete Random Variable DEFINITION: If X is a discrete random variable taking on the values x1, x2,... xk, with probabilities p1, p2,... pk, then the mean or expected value of X is given by: E( X ) x p x p x p x p X 1 1 2 2 k k i i i1 k DEFINITION: If X is a discrete random variable taking on the values x 1, x 2,... x k, with probabilities p 1, p 2,... p k, then the variance of X is given by: 2 Var( X ) X i E X x p E X E X x p and the standard deviation of X is given by: 2 2 2 2 2 2 i i SD( X ) X 2 X i 6
Let's Do It! Sum of Pips Revisited Consider the game called craps in which two fair dice are rolled. Let X be the random variable corresponding to the sum of the two dice. Its probability distribution is given below: Va l u e o f X X = x Pro b a b i l i ty P[X = x ] 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 5 4 3 2 1 Calculate the mean of X, the expected sum of the values on the two dice. You provided a graph of this distribution in let s Do It! 4. Is your expected value consistent with the idea of being the balancing point of the probability stick graph or histogram? What is the standard deviation (variability) of random variable X? 7
Let's Do It! Profits and Weather A concert promoter has a decision to make about location for a concert for the next fall. The concert can be held in an indoor auditorium or a larger facility outdoors. The following are profit forecasts, recorded in thousands of dollars, for each facility given different weather conditions: Weather Profit Indoors Profit Outdoors Probability Sunny and Warm 50 80 0.5 Sunny and Cold 50 60 0.2 Rainy and Warm 60 40 0.2 Rainy and Cold 40 6 0.1 (a) The expected profit for the indoor facility is $51,000. Compute the expected profit associated with the outdoor facility. Outdoor facility expected profit: (include your units) (b) Which location is preferred if the objective is maximizing the expected profit? (c) Which location is preferred if the promoter MUST clear a profit of at least $40,000? (d) Which location has the least variability in the profits? Explain. No calculations are necessary -- just picture the two distributions graphically. Let's Do It! Prizes One thousand tickets are sold at $1 each for prizes of $100, $50, $25, and $10. After each prize drawing, the wining ticket is then returned to the pool of tickets, what is the expected value if a person purchases two tickets? 8
At the beginning of Deal or No Deal, the contestant is presented with 26 suitcases that contain the amounts shown in the image. 1. If no cases have been opened, what is the approximate expected value? 2. Let s say that you got unlucky and blew off the 13 most valuable cases on your first 13 suitcase removals. It should be abundantly clear at this point that you re not going to walk out with a wad of cash, but what should you still be expecting? 3. If you made it to this particular point in the game and the banker were to offer you $200, would it be a statistical mistake not to accept this offer? On one night, here was the contestant s scenario: She had 5 suitcases left that contained the following amounts: $100, $400, $1000, $50,000, and $300,000; and the banker offered her $80,000 to quit. Deal or no deal? 9
Binomial Random Variables Combinations How many subsets of S 1, 2 are there which contain...... exactly 0 values (no values)? Answer : The subsets are: Combination : 1 2 1 0 2 2 2 1, 2 1 2... exactly 1 value? 2 1,... exactly 2 values? 1 1 2 Combinations n choose x represents the number of ways of selecting x items (without replacement) from a set of n distinguishable items when the order of the selection is not important is given by: n x n!! x!, where n! nn 1 n 2 21 x n Note: By definition we have 0! 1 With the TI: and n x 0 if x 0 or n x. The answer is 20 2 20! 2! 18! 190 10
Let's Do It! 5 Arrangements The Mathematics Department has a hallway with a total of ten offices for faculty. There are 3 female faculty members and 7 male faculty members that need to be assigned to these ten offices. 1 2 3 4 5 6 7 8 9 10 (a) How many possible ways are there to select the 3 offices for the female faculty from these 10 offices? (a) Give four possible selections by shading in the offices that would be assigned to the female faculty. 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 11
DEFINITION: A dichotomous or Bernoulli random variable is one which has exactly two possible outcomes, often referred to as success and failure. In this text we will only consider such variables in which the success probability p remains the same if the random experiment were repeated under identical conditions. Let's Do It! 6 Probability of a Success (a) At a local community college there are 500 freshmen enrolled, 274 sophomores enrolled, 191 juniors enrolled, and 154 seniors enrolled. An enrolled student is to be selected at random. If a success is defined to be senior, what is the probability of a success? p = (b) A standard deck of cards contain 52 cards, 13 cards of each of 4 suits. The four suits are spades, hearts, diamonds, and clubs. Each suit consists of 4 face cards (jack, queen, king, ace) and 9 numbered cards (2 through 10). A card is drawn from a well-shuffled standard deck of cards. If success is defined to be getting a face card, what is the probability of a success? p = (c) A game consists of rolling two fair die. If success is defined to be getting doubles, what is the probability of a success? p = 12
DEFINITION: A binomial random variable X is the total number of successes in n independent Bernoulli trials where on each trial the probability of a success is p. Basic Properties of a Binomial Experiment The experiment consists of n identical trials. Each trial has two possible outcomes (success,failure). The trials are independent. The probability of a success, p, remains the same for each trial. The probability of a failure is q = 1-p. The binomial random variable X is the number of successes in the n trials; X is said to have a binomial distribution denoted by Bin( n, p ); X can take on the values 0, 1, 2,..., n. Let s do it! Determine which of the following is a binomial experiment? Explain. a. Rolling a die 13 times and observing the number of spots. b. Rolling a die 13 times and observing whether the number obtained is even. c. Selecting 50 households from New York City and observing whether or not they own stocks when it is known that 28% of all households in New York own stocks. d. The probability of obtaining a head when tossing an unfair coin is 0.06. A coin is tossed until a head is obtained. 13
The Binomial Probability Distribution p n x n x x PX x p q, x 0,1,2,, n where p = probability of a success on each single trial q = 1 - p n = number of independent trials x = number of successes in the n trials x Mean, Variance, and Standard Deviation for a Binomial Random Variable Mean: E X np Variance: Var X Standard Deviation: SD X npq npq 14
Using the TI: The TI steps for finding the probability of getting exactly three successes, that is P X 3 and for finding the probability at most 2 successes, respectively, are: 15
Let's Do It! Jury Decision In a jury trial there are 12 jurors. In order for a defendant to be convicted, at least 8 of the 12 jurors must vote guilty. Assume that the 12 jurors act independently (how one juror votes will not influence how any other juror votes). Also assume that for each juror the probability that they vote correctly is 0.85. If the defendant is actually guilty, what is the probability that the jury will render a correct decision? Identify the following: A trial = n = number of independent trials = p = probability of a success on each single trial = x = number of successes in the n trials P correct decision by jury P X What is the mean and the standard deviation of the this process? 16
Let s Do It! College Drinking According to 2001 study of college students by Harvard University s School of Public Health 20% of those included in the study abstain from drinking. A random sample of six college students is selected. Identify the following: A trial = n = number of independent trials = p = probability of a success on each single trial = x = number of successes in the n trials a. What is the probability that exactly three students in this sample abstain from drinking? b. What is the probability that at most two students in this sample abstain from drinking? c. What is the probability that less than two students in this sample abstain from drinking? d. What is the probability that at least three students in this sample abstain from drinking? e. What is the probability that more than three students in this sample abstain from drinking? f. What is the mean and standard deviation of the number of students abstaining from drinking? 17
Chapter 4 Objectives: Identify Types of Quantitative Variable: (Continuous /Discrete). Determine if a table/ function is a probability mass function of the discrete random variable by checking the two assumptions : -The assigned probabilities must be between 0 and 1 and - The assigned probabilities must add up to 1. Display the Probability mass function graphically (bars), and can compute probabilities using the distribution table. Compute the Expectation (mean) and Variance/Standard deviation of a discrete random variable distribution table. Identify a binomial experiment, a trial, and probability of success/failure of the binomial process. Compute the Binomial probability in various types (Exactly=value, less than, more than value, at least a value, at most a value) by Formula/ TIcalculator. Compute the Expectation (mean) and Variance/Standard deviation of a Binomial Distribution. 18