Chapter 3: Probability Distributions and Statistics

Transcription

1 Chapter 3: Probability Distributions and Statistics Section Random Variables and Histograms A is a rule that assigns precisely one real number to each outcome of an experiment. We usually denote a random variable by X. Notes: The assignments of numbers to the outcomes of experiments are normally done in a manner that is reasonable and, most importantly, in a manner that permits these numbers to be used for interpretation and comparison. Unless otherwise specified, when the outcomes of an experiment are themselves numbers, the random variable is the rule that assigns each number to itself. Example The following are examples of random variables:.) You flip a coin. Assign the random variable X the value if heads and 2 if tails. 2.) You flip a coin 0 times. X is the number of heads that occur. 3.) You roll a die. X is the number that appears on the uppermost face of the die. Types of Random Variables: () A random variable is if it assumes only a finite number of values. (2) A random variable is if it takes on an infinite number of values that can be listed in a sequence, so that there is a first one, a second one, a third one, and so on. (3) A random variable is said to be if it can take any of the infinite number of values in some interval of real numbers.

2 Example 2 Classify the following random variables. a) X = The number of defective batteries in a sample of 8 b) X = The number of times you flip a coin until a heads is shown c) X = The amount of time an Aggie student stands during a home football game A probability distribution of a random variable assigns a probability, p, p 2,..., p n to each value of the random variable, x, x 2,..., x n. Recall that these can be represented by probability distribution tables. The probability distribution of a random variable X satisfies:. 0 p i 2. p + p p n = Example 3 Determine whether the table gives the probability distribution of the random variable X. x P (X = x) Example 4 The probability distribution of the random variable X is shown in the table. Find: a) P (X = 3) x P (X = x)

3 b) P (X 2) c) P ( 3 X 0) d) P (X > 0) Example 5 Two fair four-sided dice are rolled. Let the random variable X denote the sum of the faces that fall uppermost. a) List the outcomes of the experiment. b) Find the value assigned to each outcome of the experiment by the random variable X. c) Find the probability distribution of X. d) What is P (X 4)? 3

4 e) What is P (3 < X 6)? Example 6 A carton contains a dozen eggs, of which 5 are cracked. An experiment consists of randomly selecting 3 eggs from the carton. Let X be the number of cracked eggs that are selected. a) Find the probability distribution for X. b) What is the probability that more than cracked egg will be selected? 4

5 Example 7 An unfair die has the property that the probability of rolling a is 0.3. This die is rolled 4 times. Let X be the number of times a is rolled. Find the probability distribution for X. Note: This is a binomial experiment (section 2.4). Therefore, this distribution is referred to as a binomial distribution. We can represent a probability distribution graphically using a. The x-axis of a histogram corresponds to the values of the random variable and the y-axis corresponds to probabilities. Normally, we draw a bar of width one above each value of the random variable and the height of the bar is the probability that the random variable takes on that particular value. Area and Probability: The of a region of a histogram associated with the random variable X is equal to P (X), the probability that X occurs. Furthermore, the probability that X takes on the values in the range X i X X j is the sum of the areas of the histogram from X i to X j. Example 8 Draw a histogram for the probability distribution below. representing P (0 < X 3) and calculate its value. Shade the area X Probability

6 3.2 Measures of Central Tendency The or of the n numbers x, x 2,..., x n, denoted by µ or x, is given by Example 9 Find the mean of the following set of numbers: 64, 67, 68, 7 Example 0 Find the mean of the following set of numbers: 2, 64, 67, 68, 7 The mean is not always the best representation of what outcomes are occurring most of the time because it is sensitive to extreme values. We will look at some other notions of central tendency. The of a set of numerical data is the middle number when the numbers are arranged in order of size and there is an odd number of entries in the set. In the case that the number of entries in the set is even, the median is the mean of the two middle numbers. The of a set of observations is the observation that occurs more frequently than the others. If the frequency of occurence of two observations is the same and also greater than the frequency of occurrence of all the other observations, then we say that the set is bimodal and has two modes. If no one or two observations occurs more frequently than the others, we say that the set has no mode. Example What are the mean, median, and mode of the set of numbers:, 4, 4, 5, 9, 9, 9? (Round your answers to four decimal places.) 6

7 Example 2 What are the mean, median, and mode of the set of numbers: 5, 9, 4,, 5,? (Round your answers to four decimal places.) Finding Mean and Median Using Your Calculator Given a List of Data: () Press STAT. (2) Select EDIT. (3) In the first column, enter the values. (4) Exit to the home screen. (5) Press STAT. (6) Scroll to the right to CALC. (7) Select -Var Stats (Option ) (8) Press L (2nd ). (9) Press ENTER. (0) The mean is given by x. To find the median, scroll down until you see Med. Example 3 The daily maximum temperature in degrees Fahrenheit for the month of October in College Station follow: 85, 82, 85, 90, 89, 77, 63, 70, 85, 89, 89, 89, 86, 87, 82, 70, 90, 79, 82, 86, 90, 88, 86, 87, 88, 62, 63, 67, 7, 75, 84 Find the mean temperature and the median temperature. (Round to four decimal places.) 7

8 Finding Mean and Median Using Your Calculator Given the Random Variable and the Frequency or Probability: () Press STAT. (2) Select EDIT. (3) In the first column, enter the values the random variable X takes on. (4) In the second column, enter the frequency or probability for each value of X. (5) Exit to the home screen. (6) Press STAT. (7) Scroll to the right to CALC. (8) Select -Var Stats (Option ) (9) Press L (2nd ). Press the comma key. Press L2 (2nd 2). (0) Press ENTER. () The mean is given by x. To find the median, scroll down until you see Med. Example 4 A certain puzzle manufacturer makes and sells puzzles. The data below shows how many puzzles this company makes with different numbers of pieces. Number of Puzzles Number of Pieces a) Determine which row represents the random variable X and which row represents the frequency. b) Calculate the mean. (Round to four decimal places.) Let X denote the random variable that has values x, x 2,... x n, and let the associated probabilities be p, p 2,..., p n. The or of the random variable X, denoted by E(X) or µ, is 8

9 Example 5 Calculate the mean of the above table using our new definition by hand. (Round to four decimal places.) Example 6 Find the expected value of a random variable X having the following probability distribution. Also find the mode(s). X Probability /2 /3 /4 /6 / X 9

10 Example 7 Two cards are selected at random from a standard deck of cards. What is the expected number of hearts? Expected values are often used in games to determine whether the game is fair. In a game situation, we let X be the net winnings (profit) of the player. A game is considered fair when the expected net winnings are 0, ie, when E(X) = 0. Example 8 Visitors at a carnival pay $.00 to play a game. The game consists of pulling a marble out of a bag. If the marble is purple, the participant wins $5.00. If the marble is maroon, the participant wins $2.00. If the marble is teal, the participant wins $.00. However, if the marble is white, the participant loses. Assume the bag contains purple marble, 2 maroon marbles, 3 teal marbles, and 4 white marbles. a) Draw a histogram of the probability distribution for this game where X represents your net winnings. b) Determine the expected winnings. 0

11 c) Is this game fair? Example 9 (From Tan) In a lottery, 5000 tickets are sold for $ each. One first prize of $2000, one second prize of $500, three third prizes of $00 and ten consolation prizes of $25 are to be awarded. What are the expected net earnings of a person who buys a ticket? Example 20 A game consists of rolling a pair of fair 6-sided dice. The game costs $4 to play. If you roll the same number on both dice (a double), you win $a. Otherwise you win nothing. What value of a would make this game fair? Example 2 A life insurance company charges customers a yearly premium of $40 for a $25,000 life insurance policy. If the probability that a given customer will live through the next year is 0.998, what is the life insurance company s net earnings?

12 If the probability that the man lives drops to 0.98, what is the minimum amount of money he can expect to pay for his policy? The expected value of the binomial distribution with n trials and probability of success p in a single trial and q = p is Example 22 Given that a binomial trial is repeated 300 times with a probability of success of 0.68, find the expected value of this binomial trial. Example 23 Assume that the expected number of successes in a binomial experiment consisting of 60 trials is 23. What is the probability of success? 2

13 3.3 Measures of Spread Example 24 The following lists are test scores for 0 students in Math 66. Calculate the mean of each. a) 44,48,55,56,58,95,97,98,99,00 b) 7,72,72,74,75,75,77,77,78,79 Goal: Measure the extent of the dispersion of the data from the mean. Let X denote the random variable that takes on the values x, x 2,..., x n, and let the associated probabilities be p, p 2,..., p n. of the random variable, de- Then if µ = E(X), the noted by V ar(x), is The, denoted by σ(x) is Example 25 Compare the variances of the following distributions. A B 3

14 Finding Standard Deviation (and Variance) Using Your Calculator:. Press STAT, then EDIT 2. In the first column, enter the values the random variable X takes on. 3. In the second column, enter the frequency or probability for each value of X. 4. Exit to the home screen. 5. Press STAT. Scroll to the right to CALC. Select -Var Stats (Option ). Press L (2nd ). Press the comma key. Press L 2 (2nd 2). Press ENTER. 6. The standard deviation is given by σx. 7. To find the variance, press VARS. Select Statistics... (Option 5). Select σx (Option 4). Press x 2. Press ENTER. Example 26 Find the variance and standard deviation of the random variable X having the following probability distribution. X Probability Example 27 A certain puzzle manufacturer makes and sells puzzles. The data below shows how many puzzles this company makes with different numbers of pieces. Number of Pieces Number of Puzzles Find the standard deviation and variance for the number of pieces in a puzzle. 4