Math 166, Fall 2011,cBenjamin Aurispa3.1 Random Variables and HistogramsA random variable is a rule that assigns a number to each outcome of an experiment. We usually denotea random variable by X.There are 3 types of random variables: finite discrete, infi nite discrete, and continuous. We will defin e eachof them below.We can find probability distributions of random variables the same way we did with experiments. List allthe values that are possible for X and calculate their probabilities using any method we know.Example: A coin is tossed three times and the sequence of heads and tails is observed.1. List the outcomes of the experiment.2. Let the random variable X den ote the number of tails that occur. What are the possible values of X?3. Find th e prob ab ility distribu tion of X.4. What is P (1 ≤ X ≤ 2)?5. What is P (X > 0)?The above random variable is called finite discrete because it takes on only finitely many values.1Math 166, Fall 2011,cBenjamin AurispaConsider the experiment of rolling 2 fair 5-sided dice. We know that the sample space of this experiment isS =(1, 1), (1, 2), (1, 3), (1, 4), (1, 5),(2, 1), (2, 2), (2, 3), (2, 4), (2, 5),(3, 1), (3, 2), (3, 3), (3, 4), (3, 5),(4, 1), (4, 2), (4, 3), (4, 4), (4, 5),(5, 1), (5, 2), (5, 3), (5, 4), (5, 5)Let X be the sum of the numbers rolled.1. What are the possible values that X can take on?Since X can only be a finite number of values, X is finite discrete.2. Find th e prob ab ility distribu tion of X.3. What is P (X ≤ 8)?Example: A carton contains a dozen eggs, of which 5 are cracked. An experiment consists of randomlyselecting 3 eggs from the carton. Let X be the number of cracked eggs that are selected.1. Find th e prob ab ility distribu tion f or X.2. What is the probability that more than 1 cracked egg will be selected?2Math 166, Fall 2011,cBenjamin AurispaExample: An unfair die has the property that the probability of r olling a 1 is 0.3. This die is rolled 4 times.Let X be the number of times a 1 is rolled. Find the probability distr ibution for X.Example: A survey was conducted of families to determine the distribution of families by size. The resultsare:Fa mily Size 2 3 4 5 6Number of Families 29 16 24 11 8Let the random variable X be the number of people in a randomly chosen family. Find the pr ob ab ilitydistribution for X.Example: A die is rolled. Let the random variable X denote the number of times the die is rolled until a 5appears. What are the values that X may assume?This random variable is called infinite discrete because there are an in finite number of possible valuesand they can be arranged in a sequence of separated numbers. (We can list them all in a certain order eventhough there are infinitely many of them.)Example: Suppose I have a random variable X = The distance a person travels to work in miles. Whatvalues may X take on?This random variable is called continuous because there are infinitely many values that make up an intervalof numbers. Continuous r an dom variables often deal with time, distance, and measurements.3Math 166, Fall 2011,cBenjamin AurispaExamples: List the possible values of X and determine whether the random variable is finite discrete, infinitediscrete, or continuous.1. A marble is pulled from a jar with 3 red, 2 blue, and 3 green marbles without replacement until agreen marble is drawn. Let X be the number of number of pulls needed.2. A marble is pulled from a jar with 3 red, 2 blue, and 3 green marbles with replacement until a bluemarble is drawn. Let X be the number of number of pulls needed.3. Four marbles are pulled one at a time with replacement from a jar with 3 red, 2 blue, and 3 greenmarbles. Let X be the number of times a blue marble is pulled.4. Same experiment as above but without replacement.5. Let X be the amount of time in minutes a student works on homework on a given day.6. Let X be the number of times a 3 is rolled in 5 r olls of a die.4Math 166, Fall 2011,cBenjamin AurispaA histogram is a graphical way of displaying the probability distribution of a random variable.Plot the values of the rand om variable along the x-axis. Then draw a rectangle of width 1 centered aboveeach value with height equal to the prob ab ility of th e random variable attaining this value. (So, eachrectangle has area equal to the probability.)Remember that probability distribution values must add up to 1, which means the heights of all the rectangles(and also their areas) must add u p to 1.Draw a histogram for the probability distribution below. Shade the area representing P (0 < X ≤ 3) andcalculate its value.X 0 1 2 3 4 5Probability 0.2 0.15 0.3 0.1 0.05 0.23.2 Measures of Central TendencyThe average, or mean, of a set of numbers x1, x2, . . . , xn, denoted by x, or µ isµ =x =x1+ x2+ . . . + xnnThe media n of a set of numbers is the middle number when they are arranged in increasing order. If ther eare an even number of entries, there is no middle number, so we let the median be the average of the middletwo numbers.The mode of a s et of numbers is the number that occurs most frequently.What are the mean, median, and mode of the set of numbers 1, 4, 4, 5, 9, 9, 9?What are the mean, median, and mode of the set of numbers 1, 1, 4, 5, 5, 9?5Math 166, Fall 2011,cBenjamin AurispaYou can find m ean an d median on your calculator as follows: (Note: The mode is not given on the calculator.)1. Press STAT, 1:Edit2. Enter in the list of values u nder L1.3. Press STAT, cursor right to CALC, and then press 1: 1-Var Stats4. Then, you need to add L1, (2nd-1), so th at the home scr een reads “1-Var Stats L1”.5. Press enter. x is the mean or average. Scroll down to find the median, which is Med.Example: The daily average temperatures were recorded for the city of College Station du ring a span of twoweeks in the month of January. The results were: (I m ad e these up.)70, 67, 75, 77, 77, 69, 68, 68, 75, 68, 42, 5 5, 65, 69Find th e mean, median, and mode for this data.If data values are listed with frequencies, put the data values under L1 and their frequencies under L2.Then, type “1-Var Stats L1, L2” and press enter. By default, if you just type “1-Var Stats,” then thecalculator assumes you mean “1-Var Stats L1.”Example: A certain puzzle manufacturer makes and sells puzzles. The data below shows how many puzzlesthis company makes with d ifferent numbers of pieces.Number of Pieces