Section 6.2 - Random Variables According to the Bureau of the Census, the latest family data pertaining to family size for a small midwestern town, Nomore, is shown in Table 6.. If a family from this town is selected at random, then what is the probability of selecting a 6-person family? Because the family is being selected at random, and the size of the family can vary from 2 to 7, then family size is said to be a random variable. The proportion or percentage which is associated with each family size is interpreted as the probability of each value of family size. So the probability of choosing a family of size 3 is equal to the percentage of families of size 3. Which is? A random variable is a variable that takes on different numerical values which are determined by chance. In the previous example, family size is the random variable. Size varies among families, and the experiment involves a family being chosen randomly. Example 6. pg. 279 For each random experiment, define a random variable and identify the possible values of the variable. a. It is assumed that the largest number of children is 6, count the number of children within a family. b. Select a radial tire from the production line and determine the life of the tire, assuming no tire has ever lasted less than 20,000 miles or more than 85,000 miles. c. Count the number of heads in two tossed of a fair coin.
A random variable can be classified as either a discrete or continuous random variable depending upon the numerical values that it can assume. The number of children in a family is an example of a discrete random variable because the values of this variable: 0,, 2, 3, 4, 5, and 6 are finite or can be counted. The life of the tire is an example of a continuous random variable because the values of this variable: 20,000 to 85,000 miles can assume any value or an uncountable number of values between any two possible values of the variable. A discrete random variable is a random variable that can take on a finite or countable number of values. A continuous random variable is continuous if the value of the random variable can assume any value or an uncountable number of values between any two possible values of the variable. Review Example 6.2 pg. 280 6.3 Probability Distribution of a Discrete Random Variable A probability distribution is a distribution which displays the probabilities associated with all the possible values of a random variable. Characteristics of a Probability Distribution of a Discrete Random Variable. The probability associated with a particular value of a discrete random variable of a probability distribution is always a number between 0 and inclusive. 2. The sum of all the probabilities of a probability distribution must always be equal to one. 6.4 Mean and Standard Deviation of a Discrete Random Variable The mean value of a probability distribution is the balance point which takes into account the weights or the probability for each value of the random variable. Thus, we will refer to the mean value for a discrete random variable as a weighted mean. The mean value signifies the average number that can be expected in the long-run. 2
Example 6.7 on pg. 295 in the Text The probability distribution given in table below (Table 6.2) represents the number of computer systems a salesman named Hal expects to sell during a particular month. Probability Distribution of Number of Computer System Sold What is the random variable? Number of Computer Systems Sold X What are the possible values of the variable? What is the sum of the probabilities of this random variable? Probability P(X) 0 0.0 0.25 2 0.30 3 0.20 4 0.0 5 0.05 Now find: a. The most likely number of computer systems that Hal will sell during the month. b. The average monthly number of computer systems that Hal expects to sell. How would you interpret this result? c. The standard deviation of this probability distribution. d. The probability that the number of computer systems that Hal sells will be within one standard deviation from the mean. We will use the calculator to calculate the mean and standard deviation for a probability distribution as follows: Enter all values of the random variable, X in List Enter the probabilities, P(X) in List 2 Go to STAT and choose -Var Stats then ENTER Then type in L, L 2 (like this: -Var Stats L, L 2) See pg. 296-297 in the textbook too! 3
6.5 Binomial Probability Distribution A binomial experiment satisfies the following four conditions:. There are n identical trials. A binomial distribution is the result of a probability experiment that has been repeated a predetermined number of n times, and each repetition (trial) of the experiment is identical - Such as tossing a coin twenty times 2. The n identical trials are independent. Each outcome (trial) is independent and mutually exclusive o For example: In the experiment of tossing a coin n times, the outcome of each toss (trial) is independent of any other toss 3. The outcome for each trial can be classified as either a success or a failure. Each outcome is classified in one of two possibilities, success or fail o The determination as to whether an outcome is a success or failure is a function on how the question is asked o Success generally means a positive response to the question the toss of a coin is either a head or a tail the selection of a possible answer for a question on a multiple choice test is either correct or incorrect the toss of a die results in an outcome which is either a 5 or not a 5 a new drug will either be effective or not effective 4. The probability of a success is the same for each trial. The probability of success is the same for each trial o Meaning, in a coin tossing experiment, the probability of landing on Heads is the same for each toss (trial) of the coin Binomial Probability Formula For a binomial experiment, the probability of getting s successes in n trials is computed using the binomial probability formula. This formula is written as: s ( n s) P ( s successes in n trials) = ncs p q where: n= number of independent trials s = number of successes (n s) = number of failures nc s = the number of ways s successes can occur in n trials p = the probability of a success for one trial q = the probability of a failure for one trial = p *WE WILL USE the built-in functions of your TI83/84 calculator: binompdf or binomcdf 2 nd DISTR binompdf ENTER OR 2 nd DISTR binomcdf ENTER 4
Calculator Instructions for the BINOMIAL Distribution Summary: binompdf vs. binomcdf commands Here are some useful applications of the binomcdf and binomcdf commands: To find P(x = s), use binompdf(n, p, s) To find P(x s), use binomcdf(n, p, s) To find P(x < s), use binomcdf(n, p, s-) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ To find P(x > s), use -binomcdf(n, p, s) To find P(x s), use -binomcdf(n, p, s-) Note: s refers to the number of successes between 0 and n. If we consider the numbers from 0 to 0, what numbers do the following statements represent and write the statement as an inequality? More than 4 Less than 4 At most 4 At least 4 5
Example (NOT in text) A fair coin is tossed 0 times. Calculate the probability of: a. Getting three tails b. Getting at most one tail First state what is a success for this question (in words) Now decide what is the probability for a single success (this is the p) How many times/people are we doing it for (this is the n) For each part of a question, the number of successes (the values for s) changes n = number of independent trials = p = the probability of a success for one trial = Different for each part: s = number of successes a. Probability of getting three tails means exactly 3 tails s = 3 binompdf 0,, 3 = 0.72 binompdf ( n, p, s ) = 2 b. Probability of getting at most one tail means getting tail or less means 0 tails and tail or x 0,, 0 binompdf 0,, = 0.008 binompdf : binompdf 2 + 2 binomcdf: binompdf n, p, s = 0, 0.5, binompdf 0.008 Example 6.5 pg. 304 Each year the FBI reports the probability of a car being stolen. In a recent report, the FBI states that the probability a new car will be stolen during the year is out of 75. If you and your three friends own new cars, what is the probability that none of these cars will be stolen this year? Example 6.6 pg.304 A student is going to guess at the answers to all questions on a five question multiple choice test where there are four choices for each question. Calculate the probability of: a. Guessing three correct answers b. Guessing five correct answers c. Guessing at most two correct answers d. Guessing at least four correct answers Review Examples 6.0 and 6. on pg. 30 and Example 6.7 pg. 309 6