3-1 Discrete Random Variables

3-1 Discrete Random Variables

Example 3-1
3-2 Probability Distributions and Probability Mass Functions


Figure 3-1 Probability distribution for bits in error.
3-2 Probability Distributions and Probability Mass Functions


Figure 3-2 Loadings at discrete points on a long, thin beam.
3-2 Probability Distributions and Probability Mass Functions
Definition
Example 3-5
Example 3-5 (continued)
3-3 Cumulative Distribution Functions
Definition
Example 3-8
Example 3-8
Figure 3-4 Cumulative distribution function for Example 3-8.
3-4 Mean and Variance of a Discrete Random Variable
Definition
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.
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.

Example 3-11
3-4 Mean and Variance of a Discrete Random Variable
Expected Value of a Function of a Discrete Random Variable
3-5 Discrete Uniform Distribution
Definition
3-5 Discrete Uniform Distribution

Example 3-13
3-5 Discrete Uniform Distribution

Figure 3-7 Probability mass function for a discrete uniform random variable.
3-5 Discrete Uniform Distribution
Mean and Variance
3-6 Binomial Distribution

Random experiments and random variables
3-6 Binomial Distribution

Random experiments and random variables
3-6 Binomial Distribution

Definition
3-6 Binomial Distribution

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

Example 3-18
3-6 Binomial Distribution

Example 3-18
3-6 Binomial Distribution

Mean and Variance
3-6 Binomial Distribution

Example 3-19
3-7 Geometric and Negative Binomial Distributions

Example 3-20
3-7 Geometric and Negative Binomial Distributions

Definition
3-7 Geometric and Negative Binomial Distributions

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

3-7.1 Geometric Distribution
Example 3-21
3-7 Geometric and Negative Binomial Distributions

Definition
3-7 Geometric and Negative Binomial Distributions

Lack of Memory Property
3-7 Geometric and Negative Binomial Distributions

3-7.2 Negative Binomial Distribution
3-7 Geometric and Negative Binomial Distributions

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

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

3-7.2 Negative Binomial Distribution
3-7 Geometric and Negative Binomial Distributions

Example 3-25
3-7 Geometric and Negative Binomial Distributions

Example 3-25
3-8 Hypergeometric Distribution

Definition
3-8 Hypergeometric Distribution

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

Example 3-27
3-8 Hypergeometric Distribution

Example 3-27
3-8 Hypergeometric Distribution

Mean and Variance
3-8 Hypergeometric Distribution

Finite Population Correction Factor
3-8 Hypergeometric Distribution

Figure 3-13. Comparison of hypergeometric and binomial distributions.
3-9 Poisson Distribution

Example 3-30 
3-9 Poisson Distribution

Definition 
3-9 Poisson Distribution

Consistent Units 
3-9 Poisson Distribution

Example 3-33 
3-9 Poisson Distribution

Example 3-33 
3-9 Poisson Distribution

Mean and Variance