CHAPTER OUTLINE

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5-1.4 Independence

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Joint Probability Distributions

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5-1 TWO DISCRETE RANDOM VARIABLES 5-1.1 Joint Probability Distributions 5-1.2 Marginal Probability Distributions 5-1.3 Conditional Probability Distributions 5-2 MULTIPLE DISCRETE RANDOM VARIABLES 5-2.1 Joint Probability Distributions 5-2.2 Multinomial Probability Distribution 5-3 TWO CONTINUOUS RANDOM VARIABLES 5-3.1 Joint Probability Distributions 5-3.2 Marginal Probability Distributions

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5-3.3 Conditional Probability Distributions 5-3.4 Independence 5-4 MULTIPLE CONTINUOUS RANDOM VARIABLES 5-5 COVARIANCE AND CORRELATION 5-6 BIVARIATE NORMAL DISTRIBUTION 5-7 LINEAR COMBINATIONS OF RANDOM VARIABLES 5-8 FUNCTIONS OF RANDOM VARIABLES (CD ONLY) 5-9 MOMENT GENERATING FUNCTIONS (CD ONLY) 5-10 CHEBYSHEV S INEQUALITY (CD ONLY)

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LEARNING OBJECTIVES

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After careful study of this chapter you should be able to do the following: 1. Use joint probability mass functions and joint probability density functions to calculate probabilities 2. Calculate marginal and conditional probability distributions from joint probability distributions 3. Use the multinomial distribution to determine probabilities 4. Interpret and calculate covariances and correlations between random variables 5. Understand properties of a bivariate normal distribution and be able to draw contour plots for the probability density function 6. Calculate means and variance for linear combinations of random variables and calculate probabilities for linear combinations of normally distributed random variables

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CHAPTER 5 JOINT PROBABILITY DISTRIBUTIONS

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CD MATERIAL 7. Determine the distribution of a function of one or more random variables 8. Calculate moment generating functions and use them to determine moments for random variables and use the uniqueness property to determine the distribution of a random variable 9. Provide bounds on probabilities for arbitrary distributions based on Chebyshev s inequality Answers for most odd numbered exercises are at the end of the book. Answers to exercises whose numbers are surrounded by a box can be accessed in the e-Text by clicking on the box. Complete worked solutions to certain exercises are also available in the e-Text. These are indicated in the Answers to Selected Exercises section by a box around the exercise number. Exercises are also available for the text sections that appear on CD only. These exercises may be found within the e-Text immediately following the section they accompany.

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In s 3 and 4 we studied probability distributions for a single random variable. However, it is often useful to have more than one random variable de ned in a random experiment. For example, in the classi cation of transmitted and received signals, each signal can be classi ed as high, medium, or low quality. We might de ne the random variable X to be the number of highquality signals received and the random variable Y to be the number of low-quality signals received. In another example, the continuous random variable X can denote the length of one dimension of an injection-molded part, and the continuous random variable Y might denote the length of another dimension. We might be interested in probabilities that can be expressed in terms of both X and Y. For example, if the speci cations for X and Y are (2.95 to 3.05) and (7.60 to 7.80) millimeters, respectively, we might be interested in the probability that a part satis es both speci cations; that is, P(2.95 X 3.05 and 7.60 Y 7.80). In general, if X and Y are two random variables, the probability distribution that de nes their simultaneous behavior is called a joint probability distribution. In this chapter, we investigate some important properties of these joint distributions.

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5-1 5-1.1

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TWO DISCRETE RANDOM VARIABLES Joint Probability Distributions

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For simplicity, we begin by considering random experiments in which only two random variables are studied. In later sections, we generalize the presentation to the joint probability distribution of more than two random variables.

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EXAMPLE 5-1

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In the development of a new receiver for the transmission of digital information, each received bit is rated as acceptable, suspect, or unacceptable, depending on the quality of the received signal, with probabilities 0.9, 0.08, and 0.02, respectively. Assume that the ratings of each bit are independent. In the rst four bits transmitted, let X denote the number of acceptable bits Y denote the number of suspect bits Then, the distribution of X is binomial with n 4 and p 0.9, and the distribution of Y is binomial with n 4 and p 0.08. However, because only four bits are being rated, the possible values of X and Y are restricted to the points shown in the graph in Fig. 5-1. Although the possible values of X are 0, 1, 2, 3, or 4, if y 3, x 0 or 1. By specifying the probability of each of the points in Fig. 5-1, we specify the joint probability distribution of X and Y. Similarly to an individual random variable, we de ne the range of the random variables (X, Y ) to be the set of points (x, y) in two-dimensional space for which the probability that X x and Y y is positive.

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