By Murray Spiegel, John Schiller, Alu Srinivasan
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First, we could compute the probability distribution of X + Y, and ﬁnd the expected value and variance from there. Notice that the possible values for X + Y are 2, 3, …, 11, 12. 8333 36 36 36 CHAPTER 3: Discrete Random Variables 33 However, using Theorems 3-2 and 3-7 makes this an easy task. 5 = 7. 5) = 7 However, we could not have used Theorem 3-5 to ﬁnd the variance because we are basically using the same distribution, X, twice, and X is not independent from itself. 666 Chapter 4 CONTINUOUS RANDOM VARIABLES IN THIS CHAPTER: ✔ ✔ ✔ ✔ ✔ ✔ ✔ Continuous Random Variables Continuous Probability Distribution Distribution Functions for Continuous Random Variables Expected Values Variance Properties of Expected Values and Variances Graphical Interpretations Continuous Random Variables A nondiscrete random variable X is said to be absolutely continuous, or simply continuous, if its distribution function may be represented as 34 Copyright 2001 by the McGraw-Hill Companies, Inc.
3. 2. 333 6 6 6 The standard deviation can be found by simply taking the square root of the variance. , cm. It is for this reason that the standard deviation is often used. 30 PROBABILITY AND STATISTICS Some Theorems on Expectation Theorem 3-1: Theorem 3-2: If c is any constant, then E(cX ) = cE( X ) If X and Y are any random variables, then E ( X + Y ) = E ( X ) + E (Y ) Theorem 3-3: (11) (12) If X and Y are independent random variables, then E( XY ) = E( X ) E(Y ) (13) Note! These properties hold for any random variable, not just discrete random variables.
Examples of the Normal Distribution Since this distribution is so important, we will now run through a few examples of how to use the distribution. 3. 2. 2 is reached. Then proceed right to column marked 0. 2. 3849. 4. 21. Figure 5-2 Consider the following picture of the density curve. Figure 5-3 The required area can be broken down into two parts. 21. 46. 1772. 4864. 6636. 5. The mean weight of 500 male students at a certain college is 151 lb and the standard deviation is 15 lb. Assuming the weights are normally distributed, ﬁnd how many students weigh (a) between 120 and 155 lb, (b) more than 185 lb.
Easy Outline of Probability and Statistics by Murray Spiegel, John Schiller, Alu Srinivasan