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So if you just reserve one of the 32 bits you have at your disposal for the sign, you can have 31 bits left to store the value of the number This trade-off affects the maximum absolute value that can be stored If we reserve 1 bit for the sign, a signed integer will have 31 bits left to represent its absolute value A binary number with 31 digits can only go up to 231 1, or about 2 billion By taking one binary digit away, you halve the maximum value But wait! By adding a sign, you double the number of values that can be represented! Every number that was once an unsigned value (except zero) can now be two values: positive or negative Signed integers, instead of going as high as unsigned integers, go half as high but equally as low.

(a) Use the continuity correction to approximate the probability that 20 to 30 chips in the lot are defective (b) Use the continuity correction to approximate the probability that exactly 20 chips are defective (c) Determine the number of defective chips, x, such that the normal approximation for the probability of obtaining x defective chips is greatest..

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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The minimum value of a 32-bit unsigned integer is 0, but the minimum value of a 32-bit signed integer is 2,147,483,648 The sign bit is stored as the most signi cant bit of the 32 But in addition to this, conventional languages such as ActionScript 30 use a trick to let the CPU add two numbers the same way regardless of their sign This trick is called two s complement, and it involves counting backward in binary for the nonsign digits of negative numbers For instance, a 4-bit signed integer for 1 would be 1111, 2 would be 1110, and so on Then, adding a positive and a negative number is a normal binary addition, with the over ow ignored.

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)

I mention this because over ows in integer arithmetic are ignored, so if you are not careful with boundary cases, you can attain deceptively incorrect results as a result of an over ow You must also be aware of this when using bit math with signed integers..

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

Real numbers are represented by oating-point numbers on a computer. Just as scienti c notation can represent a huge range of numbers concisely in the form a * 10b , oating-point numbers have an incredible range, and their accuracy is proportionate to the scale of the value itself. For example, 5.12 * 10 5 is an incredibly small and precise number, 0.0000512, and 5.12 * 109 is a very large but

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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