Ch 3 KMITL 1 3 1 Discrete Random

Ch. 3 KMITL 1

3 -1 Discrete Random Variables Ch. 3 KMITL 3 -2

3 -1 Discrete Random Variables Ch. 3 KMITL 3 -3

3 -1 Discrete Random Variables: Probability Distribution Function (PDF) Ch. 3 KMITL 3 -4

3 -1 Discrete Random Variables: Cumulative Distriution Function (CDF) Ch. 3 KMITL 3 -5

3 -1 Discrete Random Variables Figure 3 -1 Probability distribution for bits in error. Ch. 3 KMITL 3 -6

3 -1 Discrete Random Variables Example 3 -4 F(x) Ch. 3 KMITL 3 -7

3 -2 Probability Distributions and Probability Mass Functions Definition Ch. 3 KMITL 3 -8

Example 3 -5 Ch. 3 KMITL 3 -9

Example 3 -5 (continued) Ch. 3 KMITL 3 -10

3 -3 Cumulative Distribution Functions Definition Ch. 3 KMITL 3 -11

Example 3 -8 Ch. 3 KMITL 3 -12

Example 3 -8 Ch. 3 KMITL 3 -13

3 -4 Mean and Variance of a Discrete Random Variable Definition Ch. 3 KMITL 3 -14

3 -4 Mean and Variance of a Discrete Random Variable Figure 3 -5 A probability distribution can be viewed as a loading with the mean equal to the balance point. Parts (a) and (b) illustrate equal means, but Part (a) illustrates a larger variance. Ch. 3 KMITL 3 -15

3 -4 Mean and Variance of a Discrete Random Variable Figure 3 -6 The probability distribution illustrated in Parts (a) and (b) differ even though they have equal means and equal variances. Ch. 3 KMITL 3 -16

3 -4 Mean and Variance of a Discrete Random Variable Ch. 3 KMITL 3 -17

3 -4 Mean and Variance of a Discrete Random Variable Ch. 3 KMITL 3 -18

3 -4 Mean and Variance of a Discrete Random Variable Ch. 3 KMITL 3 -19

3 -4 Mean and Variance of a Discrete Random Variable Ch. 3 KMITL 3 -20

Example 3 -11 Ch. 3 KMITL 3 -21

3 -4 Mean and Variance of a Discrete Random Variable Expected Value of a Function of a Discrete Random Variable Ch. 3 KMITL 3 -22

3 -4 Mean and Variance of a Discrete Random Variable Ch. 3 KMITL 3 -23

3 -5 Discrete Uniform Distribution Definition Ch. 3 KMITL 3 -24

3 -5 Discrete Uniform Distribution Example 3 -13 Ch. 3 KMITL 3 -25

3 -5 Discrete Uniform Distribution Mean and Variance Ch. 3 KMITL 3 -26

3 -6 Binomial Distribution Random experiments and random variables Ch. 3 KMITL 3 -27

3 -6 Binomial Distribution Random experiments and random variables Ch. 3 KMITL 3 -28

3 -6 Binomial Distribution Definition Ch. 3 KMITL 3 -29

3 -6 Binomial Distribution Figure 3 -8 Binomial distributions for selected values of n and p. Ch. 3 KMITL 3 -30

3 -6 Binomial Distribution Ch. 3 KMITL 3 -31

3 -6 Binomial Distribution Example 3 -16 Ch. 3 KMITL 3 -32

3 -6 Binomial Distribution Example 3 -16 Ch. 3 KMITL 3 -33

3 -6 Binomial Distribution Mean and Variance Ch. 3 KMITL 3 -34

3 -7 Geometric and Negative Binomial Distributions Example 3 -20 Ch. 3 KMITL 3 -35

3 -7 Geometric and Negative Binomial Distributions Definition Ch. 3 KMITL 3 -36

3 -7 Geometric and Negative Binomial Distributions Figure 3 -9. Geometric distributions for selected values of the parameter p. Ch. 3 KMITL 3 -37

3 -7 Geometric and Negative Binomial Distributions 3 -7. 1 Geometric Distribution Example 3 -21 Ch. 3 KMITL 3 -38

3 -7 Geometric and Negative Binomial Distributions Lack of Memory Property Ch. 3 KMITL 3 -39

3 -7 Geometric and Negative Binomial Distributions 3 -7. 2 Negative Binomial Distribution Ch. 3 KMITL 3 -40

3 -7 Geometric and Negative Binomial Distributions Figure 3 -10. Negative binomial distributions for selected values of the parameters r and p. Ch. 3 KMITL 3 -41

3 -7 Geometric and Negative Binomial Distributions Figure 3 -11. Negative binomial random variable represented as a sum of geometric random variables. Ch. 3 KMITL 3 -42

3 -7 Geometric and Negative Binomial Distributions 3 -7. 2 Negative Binomial Distribution Ch. 3 KMITL 3 -43

3 -7 Geometric and Negative Binomial Distributions Example 3 -25 Ch. 3 KMITL 3 -44

3 -7 Geometric and Negative Binomial Distributions Example 3 -25 Ch. 3 KMITL 3 -45

3 -8 Hypergeometric Distribution Definition Ch. 3 KMITL 3 -46

3 -8 Hypergeometric Distribution Figure 3 -12. Hypergeometric distributions for selected values of parameters N, K, and n. Ch. 3 KMITL 3 -47

3 -8 Hypergeometric Distribution Ch. 3 KMITL 3 -48

3 -8 Hypergeometric Distribution Example 3 -27 Ch. 3 KMITL 3 -49

3 -8 Hypergeometric Distribution Example 3 -27 Ch. 3 KMITL 3 -50

3 -8 Hypergeometric Distribution Mean and Variance Ch. 3 KMITL 3 -51

3 -8 Hypergeometric Distribution Finite Population Correction Factor Ch. 3 KMITL 3 -52

3 -8 Hypergeometric Distribution Figure 3 -13. Comparison of hypergeometric and binomial distributions. Ch. 3 KMITL 3 -53

3 -9 Poisson Distribution Definition Ch. 3 KMITL 3 -54

3 -9 Poisson Distribution Properties of the Poisson Process 1. The number of outcomes occurring in one time interval or specified region of space is independent of the number that occur in any other disjoint time interval or region. In this sense we say that the Poisson process has no memory. 2. The probability that a single outcome will occur during a very short time interval or in Ch. 3 a small region is proportional KMITL to the length 3 -55

3 -9 Poisson Distribution Poisson density functions for different means Ch. 3 KMITL 3 -56

3 -9 Poisson Distribution Example 3 -30 Ch. 3 KMITL 3 -57

3 -9 Poisson Distribution Example 3 -30 Ch. 3 KMITL 3 -58

3 -9 Poisson Distribution Consistent Units Ch. 3 KMITL 3 -59

3 -9 Poisson Distribution Ch. 3 KMITL 3 -60

3 -9 Poisson Distribution Example 3 -33 Ch. 3 KMITL 3 -61

3 -9 Poisson Distribution Example 3 -33 Ch. 3 KMITL 3 -62

3 -9 Poisson Distribution Mean and Variance Ch. 3 KMITL 3 -63