AP Statistics Ch. 6 Notes Random Variables A variable is any characteristic of an individual (remember that individuals are the objects described by a data set and may be eole, animals, or things). Variables tell us what information was gathered. A random variable is a variable that takes numerical values that describe the outcomes of some chance rocess. Examles of random variables: the height of a randomly-selected student, the number of heads in three flis of a coin, the difference between the diameters of a randomly-chosen cu and a randomly-chosen lid. The robability distribution of a random variable gives its ossible values and their robabilities. A discrete random variable takes a fixed set of ossible values with gas between them. Discrete random variables are often the result of counting something, but can also be things like shoe sizes that have gas between ossible values. The robability distribution of a discrete random variable lists the values x i and their robabilities i : Value: x x x 3 Probability: 3 The robabilities i must satisfy two requirements:. Each robability i is a number between 0 and.. The sum of the robabilities is : + + 3 +... = To find the robability of any event, add the robabilities i of the articular values x i that make u the event. When describing a robability distribution, remember to mention shae, center, and variability. Examle: In 00, there were 39 games layed in the National Hockey League s regular season. Imagine selecting one of these games at random and then randomly selecting one of the two teams that layed in the game. Define the random variable = number of goals scored by a randomly-selected team in a randomlyselected game. The table below gives the robability distribution of. # of Goals: 0 3 4 5 6 7 8 9 Probability: 0.06 0.54 0.8 0.9 0.73 0.094 0.04 0.05 0.004 0.00 a) Show that the robability distribution for is legitimate. b) Make a histogram of the robability distribution. Describe what you see. c) What is the robability that the number of goals scored by a randomly-selected team in a randomlyselected game is at least 6?
The mean μ or exected value of a discrete random variable is its average value over many, many reetitions of the chance rocess. It is calculated by taking into account the fact that the outcomes may not be equally likely. Suose that is a discrete random variable whose robability distribution is Value: x x x 3 Probability: 3 To find the mean (exected value) of, multily each ossible value by its robability, then add all the roducts: μ = E ( ) = x + x + x3 3 +... = xi i. Examle: In the revious examle, we defined the random variable to be the number of goals scored by a randomly-selected team in a randomly-selected hockey game in 00. The table below gives the robability distribution of : # of Goals: 0 3 4 5 6 7 8 9 Probability: 0.06 0.54 0.8 0.9 0.73 0.094 0.04 0.05 0.004 0.00 Calculate the mean of the random variable and interret this value in context. The standard deviation σ of a random variable is a measure of how much the values of the variable tyically vary from the mean μ. Suose that is a discrete random variable whose robability distribution is Value: x x x 3 Probability: 3 and that μ is the mean of. The variance of is ( ) = = ( ) + ( ) + ( ) + = ( ) Var σ x μ x μ x μ... x μ. 3 3 i i The standard deviation of, σ, is the square root of the variance. Examle: Calculate and interret the standard deviation of the random variable from the revious examle.
A continuous random variable takes all values in an interval of numbers. The robability distribution of is described by a density curve. The robability of any event is the area under the density curve above the values of that make u the event. All continuous robability models assign robability 0 to every individual outcome. This is because each outcome is just one of an infinite number of ossible outcomes, so the robability of each individual outcome is. A robability of P( 68 70) is really the same as P( 68 70) above 68 or 70. < < because there is no area directly Examle: The weights of three-year-old girls closely follow a Normal distribution with a mean of μ = 30.7 ounds and a standard deviation of σ = 3.6 ounds. Randomly choose one three-year-old girl and call her weight. a) Make an accurate sketch of the density curve for the weights of three-year-old girls. Label the oints,, and 3 standard deviations from the mean. b) Find the robability that the randomly-selected three-year-old girl weighs less than 5 ounds. c) Find the robability that the randomly-selected three-year-old girl weighs at least 30 ounds. d) Find the robability that the randomly-selected three-year-old girl weighs between 6 and 3 ounds. e) The robability that the randomly-selected three-year-old girl is considered underweight is 0.05. What weights are considered underweight for three-year-old girls?
Transforming and Combining Random Variables Examle: Consider the random variable = the outcome of one roll of a fair die. μ σ σ Outcome: 3 4 5 6 Probability: /6 /6 /6 /6 /6 /6 ( ) ( )... 6( ) 3.5 6 6 6 ( ) ( ) ( ) ( ) ( ) ( ) = 3.5 + 3.5 +... + 6 3.5.967 6 6 6 = + + + =.967.7 Let Y = (the result of doubling the outcome of one roll of a die.) Give the robability distribution of Y and calculate its mean, variance, and standard deviation. Let Z = + (the result of rolling two dice and adding the outcomes.) Give the robability distribution of Z and calculate its mean, variance, and standard deviation. What relationshis do you observe between the robability distributions, means, variances, and standard deviations of, Y, and Z?
Examle: El Dorado Community College considers a student to be full-time if he or she is taking between and 8 units. The number of units that a randomly-selected El Dorado Community College full-time student is taking in the fall semester has the following distribution: # of Units (): 3 4 5 6 7 8 Probability: 0.5 0.0 0.05 0.30 0.0 0.05 0.5 Here is a histogram of the robability distribution along with the mean and standard deviation of. Probability 0.30 0.5 0.0 0.5 0.0 0.05 0.00 3 4 5 6 7 8 9 Number of Units, ( 0.5) 3( 0.0 )... 8( 0.5) 4.65 ( ) ( ) ( ) ( ) ( ) ( ) μ = + + + = σ = 4.65 0.5 + 3 4.65 0.0 +... + 8 4.65 0.5 = 4.3 σ = 4.3 =.06 The tuition for full-time students is $50 er unit. That is, if T = tuition charge for a randomly-selected full-time student, T = 50. Here is the robability distribution for T, a histogram of the robability distribution, and the mean and standard deviation of T. Tuition (T): $600 $650 $700 $750 $800 $850 $900 Probability: 0.5 0.0 0.05 0.30 0.0 0.05 0.5 Probability 0.30 0.5 0.0 0.5 0.0 0.05 0.00 600 650 700 750 800 850 900 950 Tuition, T (dollars) T 600( 0.5) 650( 0.0 )... 900( 0.5 ) $73.50 ( ) ( ) ( ) ( ) ( ) ( ) μ = + + + = σ = 600 73.5 0.5 + 650 73.5 0.0 +... + 900 73.5 0.5 = 0, 568 T σ = 0,568 = $03 T In addition to tuition charges, each full-time student is assessed student fees of $00 er semester. If C = overall cost for a randomly-selected full-time student, then C = T + 00 or C = 50 + 00. Here is the robability distribution for C, a histogram of the robability distribution, and the mean and standard deviation of C. Overall Cost (C): $700 $750 $800 $850 $900 $950 $000 Probability: 0.5 0.0 0.05 0.30 0.0 0.05 0.5 Probability 0.30 0.5 0.0 0.5 0.0 0.05 0.00 700 750 800 850 900 950 000 050 Overall Cost, C (dollars) C 700( 0.5) 750( 0.0 )... 000( 0.5 ) $83.50 ( ) ( ) ( ) ( ) ( ) ( ) μ = + + + = σ = 700 83.5 0.5 + 750 83.5 0.0 +... + 000 83.5 0.5 = 0, 568 C σ = 0,568 = $03 C How do the shaes of the robability distributions of μ C comare? How do the values of σ, σ T, and σ C comare?, T, and C comare? How do the values of μ, μ, and T
Effect on a Random Variable of Multilying (or Dividing) by a Constant Multilying (or dividing) each value of a random variable by a number b : Multilies (or divides) measures of center and location (mean, median, quartiles, ercentiles) by b. Multilies (or divides) measures of sread (range, IQR, standard deviation) by b. Does not change the shae of the distribution. Effect on a Random Variable of Adding (or Subtracting) a Constant Adding the same number a (which could be negative) to each value of a random variable: Adds a to measures of center and location (mean, median, quartiles, ercentiles). Does not change shae or measures of sread (range, IQR, standard deviation). Effects of a Linear Transformation on the Mean and Standard Deviation of a Random Variable If Y = a + b is a linear transformation of the random variable, then the robability distribution of Y has the same shae as the robability distribution of. μ = a + bμ. Y σy = bσ (since b could be a negative number). Whether we re dealing with data or random variables, the effects of a linear transformation are the same! These results aly to both discrete and continuous random variables. Examle: In a large introductory statistics class, the distribution of = raw test score of a randomly-selected student was aroximately Normally distributed with a mean of 7. and a standard deviation of 3.8. The rofessor decides to scale the scores by multilying the raw scores by 4 and adding 0. a) Define the random variable Y to be the scaled test score of a randomly-selected student from this class. Find the mean and standard deviation of Y. b) What is the robability that a randomly-selected student has a scaled test score of at least 90? c) The rofessor wants the robability of failing to be 0.. What scaled score should he set as the cutoff score between assing and failing? d) The rofessor would like about 0% of students to receive A s. What scaled score should he set as the cutoff score between an A and a B?
Combining Random Variables Mean of the Sum of Random Variables For any two random variables and Y, the exected value of + Y is ( + ) = = +. E Y μ μ μ + Y Y In general, the mean of the sum of several random variables is the sum of their means. Indeendent Random Variables: If knowing whether any event involving alone has occurred tells us nothing about the occurrence of any event involving Y alone, and vice versa, then and Y are indeendent random variables. Variance of the Sum of Indeendent Random Variables (Pythagorean Theorem of Statistics) For any two indeendent random variables and Y, the variance of + Y is σ + Y = σ + σy In general the variance of the sum of several indeendent random variables is the sum of their variances. You can only add variances if the two random variables are indeendent! Standard deviations don t add always work with the variances! Examle: A large auto dealershi kees track of sales and lease agreements made during each hour of the day. Let = the number of cars sold and Y = the number of cars leased during the first hour of business on a randomly-selected Friday. Based on revious records, the robability distributions of and Y are as follows: Cars Sold: 0 3 Probability: 0.3 0.4 0. 0. μ =. σ = 0.943 Define T = + Y. Cars Leased: 0 Probability: 0.4 0.5 0. μ Y = 0.7 σ = 0.64 Y a) Find and interret μ T. b) Comute and interret σ T assuming that and Y are indeendent.
c) The dealershi s manager receives a $500 bonus for each car sold and a $300 bonus for each car leased. Find the mean and standard deviation of the manager s total bonus B. Mean of the Difference of Random Variables For any two random variables and Y, the exected value of Y is ( ) = =. E Y μ μ μ Y Y In general, the mean of the difference of several random variables is the difference of their means. Variance of the Difference of Random Variables For any two indeendent random variables and Y, the variance of Y is σ Y = σ + σy In general the variance of the sum of several indeendent random variables is the sum of their variances. Even though we are subtracting the random variables, we still add the variances. This is because more sources of variation leads to more variability, no matter how we are combining them. Examle: Refer to the revious examle. Recall that = the number of cars sold and Y = the number of cars leased during the first hour of business on a randomly-selected Friday. μ =. σ = 0.943 Define D = Y. a) Find and interret μ D. μ Y = 0.7 σ = 0.64 Y b) Comute σ D assuming that and Y are indeendent.
c) Recall that the dealershi s manager receives a $500 bonus for each car sold and a $300 bonus for each car leased. Find the mean and standard deviation of the difference in the manager s bonus for cars sold and leased. The sum or difference of indeendent Normal random variables is also Normally distributed. Examle: Suose that a certain variety of ales have weights that are aroximately Normally distributed with a mean of 9 ounces and a standard deviation of.5 ounces. If bags of ales are filled by randomly selecting ales, what is the robability that the sum of the weights of the ales is less than 00 ounces? Assume that the weights of the ales are indeendent. Examle: Suose that the height M of male seed daters follows a Normal distribution, with a mean of 70 inches and a standard deviation of 3.5 inches, and suose that the height F of female seed daters follows a Normal distribution, with a mean of 65 inches and a standard deviation of 3 inches. What is the robability that a randomly-selected male seed dater is taller than the randomly-selected female seed dater with whom he is aired?
Binomial and Geometric Random Variables When the same chance rocess is reeated several times, we are often interested in whether a articular outcome does or doesn t haen on each reetition. A binomial setting is one in which the following conditions are met: Binary: All ossible outcomes of each trial can be classified as either successes or failures. Indeendent: The trials are indeendent. Knowing the result of one trial must not tell us anything about the result of any other trial. Fixed Probability of Success: The robability of success,, is the same for each trial. Fixed Number of Trials: You are interested in the number of successes in a fixed number of trials, n. The Indeendent condition involves conditional robabilities (robability of success on the nd trial given success on the st trial). The Fixed Probability condition involves unconditional robabilities (robability of success on a trial without considering the outcome of any other trial). Get used to checking conditions! On the AP test, you will be exected to list conditions for several tyes of statistical rocedures and exlain why each of them are met (or not) before roceeding with the roblem. Binomial Random Variable: The count of successes,, in a binomial setting. The robability distribution of is called a binomial distribution. It lists all ossible numbers of successes and the robability of each number of successes. Binomial distributions can be comletely defined by giving the values of n (the number of trials) and (the robability of success on each trial). Examles: Determine whether the random variables have a binomial distribution. Justify your answer. a) Roll a fair die 0 times and let = the number of sixes. b) Shoot a basketball 0 times from various distances on the court. Let Y = the number of shots made. c) Shuffle a deck of cards. Turn over the to five cards one at a time and let H = the number of hearts. d) Roll a air of dice reeatedly until you roll doubles. Let = the number of rolls.
Examle: The robability of rolling doubles when rolling two dice is 6 36 or 6. If you are laying a game in which rolling doubles is beneficial, what is the robability that you will roll doubles on exactly of your next 5 turns? a) List all the ossible ways you could end u with doubles on exactly of your next 5 turns. How many different ways are there for that to haen? It wasn t too bad to list the ossibilities and count in this case, but what if I had asked for the robability of rolling exactly 7 doubles in your next 40 rolls? Good luck listing all the ossible ways that could haen! (There are 8,643,560 of them)! Gosh, wouldn t it be nice if there was some sort of formula we could use to find out how many ways there were to have k successes in n trials? Binomial Coefficient: The number of ways for there to be k successes in n trials is given by n n! = k k! ( n k )! n is also called a combination and is sometimes abbreviated nc k. It is read n choose k. k b) Verify that you listed all the ossible ways to roll doubles on of your next 5 turns using a combination. c) The robability of each of the secific ossibilities you listed in a) is the same. Calculate this robability. d) Combine your answers from the stes above to calculate the robability of rolling doubles on exactly of your next 5 turns. Binomial Probability: If has the binomial distribution with n trials and robability of success on each trial, then the ossible values of are 0,,,..., n. If k is any one of these values, then n k ( ) ( ) k n k n k P = k = or P ( = k ) = q, where k k n = number of trials = robability of success on each trial k = number of successes q = = robability of failure on each trial Note: I find it easier to think of this formula in words: # of trials robability of success # of successes ( ) ( robability of failure) # of successes # of failures
Binomial Probabilities on a Calculator The Distribution menu ( nd Vars) on a TI-83 or TI-84 has two commands involving binomial robabilities: binomdf (n,, k) gives P ( = k), the robability of exactly k successes in n trials. binomcdf (n,, k) gives P ( k), the robability of k or fewer successes in n trials. You may write these commands on the AP test, but you must define all numbers. For examle, you could write: P = 3 = binomialdf n = 5, = 0.4, k = 3 = 0.304 ( ) ( ) If you are interested, PDF stands for robability distribution function and CDF stands for cumulative distribution function. They can be best understood by thinking of a robability histogram or a density curve. Probability 0.0 0.5 0.0 0.05 Binomial, n=5, =0. Density 0.0 0.5 0.0 0.05 Normal, Mean=5, StDev= 0.00 0 3 4 5 6 7 8 9 0 0.00 0 4 6 8 0 The robability distribution function (PDF) gives you the y-coordinate for a certain value of. o For a discrete robability distribution (which includes binomial), this is the height of the bar of a robability histogram corresonding to that value, so it is the robability of the secific value. o For a continuous robability distribution (like a normal distribution), this number is simly the height of the density curve at that oint. It doesn t mean anything in terms of robability, since in a continuous robability distribution, the robability of each secific value is zero. The cumulative distribution function (CDF) always gives the robability that is less than or equal to a certain value. o For a discrete robability distribution, it is the total area of the bars of a histogram corresonding to the values less than or equal to the value of interest. o For a continuous robability function, the CDF is the area under the density curve to the left of the value of interest. For you calculus tyes, the CDF is the integral of the PDF from to x. Examle: A et food manufacturer is romoting a new brand with a rebate offer on its 0-ound bag. Each ackage is suosed to contain a couon for a $4.00 mail-in rebate. The comany has found that the machine disensing these couons fails to lace a couon in 0% of the bags. A dog owner buys 5 bags. Let = the number of bags bought by the dog owner that don't contain a couon. a) Check conditions to verify that is a binomial random variable. b) Find the robability that exactly one of the bags will not contain a couon.
c) Find the robability that at most two bags will fail to have a couon. Examle: Mr. Meany-Pants gives his class a 0-question multile choice quiz. Each question has 5 ossible answers. However, all of the questions and answer choices are written in Arabic, and none of the students seak Arabic, so they are forced to guess on every question. Mr. Meany-Pants used a random number generator to decide where to ut the correct answer, and each answer has an equal chance of being correct. George is one of the students in the class. Let = the number of questions George answers correctly. a) Show that is a binomial random variable. b) Find P( = 3 ). Exlain what this result means. c) To get a assing score on the quiz, a student must guess correctly on at least 6 roblems. Would you be surrised if George earned a assing score? Comute an aroriate robability to suort your answer. Mean and Standard Deviation of a Binomial Distribution Examle: Let = number of doubles in 4 rolls of two dice. is binomial with n = 4 and = 6. The robability distribution is given below. Find the mean and standard deviation of. We could use the formulas for mean and standard deviation of a robability distribution that we already know. μ = 0 0.48 + 0.386 + 0.6 + 3 0.05 + 4 0.00 = 0.667 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) σ = 0 0.667 0.48 + 0.667 0.386 + 0.667 0.6 + 3 0.667 0.05 + 4 0.667 0.00 = 0.556 σ = 0.556 = 0.746 Number of Doubles (): 0 3 4 Probability: 0.48 0.386 0.6 0.05 0.00 As fun as that was, it would be nice if we didn t have to do this every time, esecially if n is some huge number. Luckily, there is a shortcut for binomial distributions.
Mean and Standard Deviation of a Binomial Random Variable: If a count has the binomial distribution with number of trials n and each trial has robability of success, the mean and standard deviation of are μ = n These formulas work ONLY for binomial distributions! They can t be σ = n( ) or σ = nq used for any other tye of distribution. Examle: The makers of a diet cola claim that its taste is indistinguishable from the full-calorie version of the same cola. To investigate, an AP Statistics student named Sally reared small samles of each tye of cola in identical cus. Then she had 30 volunteers taste each cola in a random order and try to identify which was the diet cola and which was the regular cola. If we assume that the volunteers really couldn t tell the difference, then each one was guessing with a 50% chance of being correct. Let = the number of volunteers who correctly identify the colas. a) Exlain why is a binomial random variable. b) Find the mean and the standard deviation of. Interret each value in context. c) Of the 30 volunteers, 3 made correct identifications. Does this give convincing evidence that the volunteers can taste the difference between the diet and regular colas? Suort your answer with an aroriate robability. Binomial Distributions in Statistical Samling Examle: A drawer contains 3 blue ens and 7 black ens. I reach in and draw 5 ens at random. What is the robability that I draw exactly blue ens? Exlain why this is not a binomial setting.
Samling Without Relacement Condition (0% Condition) When taking an SRS of size n from a oulation of size N, the indeendence condition is violated. However, it is safe to use a binomial distribution to model the count of successes in the samle as long as the samle size is at most 0% of the oulation size or n N. Phrased differently, the oulation must be at least 0 times 0 as large as the samle, or N 0 n. This does not mean that small samles are better! Larger samles almost always give more conclusive results. We just can t use the binomial distribution if this condition is not met. We would use a distribution called a hyergeometric distribution, which is not art of the AP curriculum. Examle: Suose you have a bag with 8 AAA batteries but only 6 of them are good. You need to choose 4 for your grahing calculator. If you randomly select 4 batteries, what is the robability that all 4 of the batteries you 4 4 0 choose will work? Exlain why the answer isn t P ( = 4) = ( 0.75) ( 0.5) = 0.364. (The actual 4 robability is about 0.43). Examle: A survey found that business is the most oular college major for male college students in the U.S. who lay basketball or football, with 37% selecting this major. Find the robability that a random samle of 00 male college athletes in these two sorts would contain more than 80 business majors. Is it aroriate to use binomial calculations to find this robability? Justify your answer. If it is aroriate, find the robability. Geometric Random Variables A geometric setting is one in which we erform indeendent trials of the same chance rocess and record the number of trials until a articular outcome occurs. The following are the conditions are met: Binary: All ossible outcomes of each trial can be classified as either successes or failures. Indeendent: The trials are indeendent. Knowing the result of one trial must not tell us anything about the result of any other trial. Fixed Probability of Success: The robability of success,, is the same for each trial. Count # of Trials to st Success: The goal is to count the number of trials until the first success. Notice that the first three conditions are the same as those for a binomial setting. The only difference is that the goal is different. In a binomial setting, we count the number of successes in a fixed number of trials. In a geometric setting, we count the number of trials before the first success.
Geometric Probability: The number of trials Y that it takes to get a success in a geometric setting is a geometric random variable. The robability distribution of Y is a geometric distribution. A geometric distribution is comletely defined by giving the value of, the robability of success on each trial. The ossible values of Y are,, 3, If k is any one of these values, k ( ) ( ) or ( ) k P Y = k = P Y = k = q Geometric Probabilities on a Calculator The Distribution menu ( nd Vars) on a TI-83 or TI-84 has two commands involving geometric robabilities: geometdf (, k) gives P ( = k), the robability that the first success haens on the k th trial. geometcdf (, k) gives P ( k), the robability that the first success haens on or before the k th trial. Examle: Consider the random variable Y = the number of rolls of two dice that it takes to roll doubles for the first time. a) Verify that this is a geometric setting. b) Find the robability that it takes 3 turns to roll doubles. c) Find the robability that it takes more than 3 turns to roll doubles. Mean (Exected Value) of a Geometric Random Variable: If Y is a geometric random variable with robability of success on each trial, then its mean is μy = E ( Y ) = That is, the exected number of trials required to get the first success is. Examle: How many times would you exect to have to roll two dice before you roll doubles for the first time? In case you are wondering, the standard deviation of a geometric random variable is σy =.