3 6 Chapter Discrete Probability Distributions © 2010 Pearson Prentice Hall. All rights reserved

Section 6.1 Probability Rules © 2010 Pearson Prentice Hall. All rights reserved

© 2010 Pearson Prentice Hall. All rights reserved

© 2010 Pearson Prentice Hall. All rights reserved A random variable is a numerical measure of the outcome from a probability experiment, so its value is determined by chance. Random variables are denoted using letters such as X. © 2010 Pearson Prentice Hall. All rights reserved

© 2010 Pearson Prentice Hall. All rights reserved A discrete random variable has either a finite or countable number of values. The values of a discrete random variable can be plotted on a number line with space between each point. See the figure. © 2010 Pearson Prentice Hall. All rights reserved

© 2010 Pearson Prentice Hall. All rights reserved A continuous random variable has infinitely many values. The values of a continuous random variable can be plotted on a line in an uninterrupted fashion. See the figure. © 2010 Pearson Prentice Hall. All rights reserved

© 2010 Pearson Prentice Hall. All rights reserved EXAMPLE Distinguishing Between Discrete and Continuous Random Variables Determine whether the following random variables are discrete or continuous. State possible values for the random variable. The number of light bulbs that burn out in a room of 10 light bulbs in the next year. (b) The number of leaves on a randomly selected Oak tree. (c) The length of time between calls to 911. Discrete; x = 0, 1, 2, …, 10 Discrete; x = 0, 1, 2, … Continuous; t > 0 © 2010 Pearson Prentice Hall. All rights reserved

© 2010 Pearson Prentice Hall. All rights reserved

© 2010 Pearson Prentice Hall. All rights reserved A probability distribution provides the possible values of the random variable X and their corresponding probabilities. A probability distribution can be in the form of a table, graph or mathematical formula. © 2010 Pearson Prentice Hall. All rights reserved

© 2010 Pearson Prentice Hall. All rights reserved EXAMPLE A Discrete Probability Distribution The table to the right shows the probability distribution for the random variable X, where X represents the number of DVDs a person rents from a video store during a single visit. x P(x) 0.06 1 0.58 2 0.22 3 0.10 4 0.03 5 0.01 © 2010 Pearson Prentice Hall. All rights reserved

© 2010 Pearson Prentice Hall. All rights reserved

© 2010 Pearson Prentice Hall. All rights reserved EXAMPLE Identifying Probability Distributions Is the following a probability distribution? x P(x) 0.16 1 0.18 2 0.22 3 0.10 4 0.30 5 0.01 © 2010 Pearson Prentice Hall. All rights reserved

© 2010 Pearson Prentice Hall. All rights reserved EXAMPLE Identifying Probability Distributions Is the following a probability distribution? x P(x) 0.16 1 0.18 2 0.22 3 0.10 4 0.30 5 -0.01 © 2010 Pearson Prentice Hall. All rights reserved

© 2010 Pearson Prentice Hall. All rights reserved EXAMPLE Identifying Probability Distributions Is the following a probability distribution? x P(x) 0.16 1 0.18 2 0.22 3 0.10 4 0.30 5 0.04 © 2010 Pearson Prentice Hall. All rights reserved

© 2010 Pearson Prentice Hall. All rights reserved

© 2010 Pearson Prentice Hall. All rights reserved A probability histogram is a histogram in which the horizontal axis corresponds to the value of the random variable and the vertical axis represents the probability of that value of the random variable. © 2010 Pearson Prentice Hall. All rights reserved

© 2010 Pearson Prentice Hall. All rights reserved EXAMPLE Drawing a Probability Histogram x P(x) 0.06 1 0.58 2 0.22 3 0.10 4 0.03 5 0.01 Draw a probability histogram of the probability distribution to the right, which represents the number of DVDs a person rents from a video store during a single visit. © 2010 Pearson Prentice Hall. All rights reserved

© 2010 Pearson Prentice Hall. All rights reserved

© 2010 Pearson Prentice Hall. All rights reserved EXAMPLE Computing the Mean of a Discrete Random Variable Compute the mean of the probability distribution to the right, which represents the number of DVDs a person rents from a video store during a single visit. x P(x) 0.06 1 0.58 2 0.22 3 0.10 4 0.03 5 0.01 © 2010 Pearson Prentice Hall. All rights reserved

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© 2010 Pearson Prentice Hall. All rights reserved The following data represent the number of DVDs rented by 100 randomly selected customers in a single visit. Compute the mean number of DVDs rented. © 2010 Pearson Prentice Hall. All rights reserved

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© 2010 Pearson Prentice Hall. All rights reserved As the number of trials of the experiment increases, the mean number of rentals approaches the mean of the probability distribution. © 2010 Pearson Prentice Hall. All rights reserved

© 2010 Pearson Prentice Hall. All rights reserved

© 2010 Pearson Prentice Hall. All rights reserved Because the mean of a random variable represents what we would expect to happen in the long run, it is also called the expected value, E(X), of the random variable. © 2010 Pearson Prentice Hall. All rights reserved

© 2010 Pearson Prentice Hall. All rights reserved EXAMPLE Computing the Expected Value of a Discrete Random Variable A term life insurance policy will pay a beneficiary a certain sum of money upon the death of the policy holder. These policies have premiums that must be paid annually. Suppose a life insurance company sells a $250,000 one year term life insurance policy to a 49-year-old female for $530. According to the National Vital Statistics Report, Vol. 47, No. 28, the probability the female will survive the year is 0.99791. Compute the expected value of this policy to the insurance company. x P(x) 530 0.99791 530 – 250,000 = -249,470 0.00209 Survives Does not survive E(X) = 530(0.99791) + (-249,470)(0.00209) = $7.50 © 2010 Pearson Prentice Hall. All rights reserved

© 2010 Pearson Prentice Hall. All rights reserved

© 2010 Pearson Prentice Hall. All rights reserved

© 2010 Pearson Prentice Hall. All rights reserved EXAMPLE Computing the Variance and Standard Deviation of a Discrete Random Variable x P(x) 0.06 1 0.58 2 0.22 3 0.10 4 0.03 5 0.01 Compute the variance and standard deviation of the following probability distribution which represents the number of DVDs a person rents from a video store during a single visit. x P(x) 0.06 -1.43 2.0449 0.122694 1 0.58 -0.91 0.8281 0.480298 2 0.22 -1.27 1.6129 0.354838 3 0.1 -1.39 1.9321 0.19321 4 0.03 -1.46 2.1316 0.063948 5 0.01 -1.48 2.1904 0.021904 © 2010 Pearson Prentice Hall. All rights reserved