# Discrete Random Variables & Distribution

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### 3-1 Discrete Random Variables :

3-1 Discrete Random Variables

### 3-1 Discrete Random Variables :

3-1 Discrete Random Variables Example 3-1

### 3-2 Probability Distributions and Probability Mass Functions :

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 :

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 :

3-2 Probability Distributions and Probability Mass Functions Definition

Example 3-5

### Slide 9:

Example 3-5 (continued)

### 3-3 Cumulative Distribution Functions :

3-3 Cumulative Distribution Functions Definition

Example 3-8

### Slide 12:

Example 3-8 Figure 3-4 Cumulative distribution function for Example 3-8.

### 3-4 Mean and Variance of a Discrete Random Variable :

3-4 Mean and Variance of a Discrete Random Variable Definition

### 3-4 Mean and Variance of a Discrete Random Variable :

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 :

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 :

3-4 Mean and Variance of a Discrete Random Variable Expected Value of a Function of a Discrete Random Variable

### Describing dataMeasures of Central Tendency and Dispersion. :

Describing dataMeasures of Central Tendency and Dispersion. I. What is a measure of Central Tendency? Often a single number is needed to represent a set of data. Arithmetic Mean or average

### Describing dataMeasures of Central Tendency and Dispersion. :

Describing dataMeasures of Central Tendency and Dispersion. Define: Statistics A measurable characteristic of a sample. Define: Parameter A measurable characteristic of a population population mean.

### Describing dataMeasures of Central Tendency and Dispersion. :

Describing dataMeasures of Central Tendency and Dispersion. Median: properties of the Median. Mode: Define: ModeThe value of the observation that appears most frequently.

### Why study Dispersion? :

Why study Dispersion? Remark: A measure of Central Tendency is representative if data are clustered close to it. There are several reasons for analyzing the dispersion in a set of data.

### Summarizing DataFrequency Distribution and Graphic Presentation :

Summarizing DataFrequency Distribution and Graphic Presentation Goals: Organize raw data into a frequency distribution. Portray the frequency distribution in histogram a cumulative frequency. Present data using such common graphic techniques: line charts, bar chats, and pie charts.

### Frequency Distribution :

Frequency Distribution Define: A grouping of data into categories showing the number of observation in each mutually exclusive category Determining class interval: Suggesting class interval = A small value indicates that the data are clustered closely: The mean is a representative of the data set. The mean is a reliable average. A large value means the mean is not reliable. To compare the spread in two or more distribution.

### Measures of dispersion :

Measures of dispersion Range: the difference between the highest value and lowest value. Mean Deviation (MAD)

Mean Deviation (MAD) Advantage and Disadvantage of MAD Two advantages: It uses the value of every item in a set of data It's the mean amount by which the value deviate from the mean. Disadvantage: Absolute value are difficult to calculate

### Measures of dispersion :

Measures of dispersion Variance and Standard deviation. Sample variance: Sample Standard Deviation:

### Box-Plots :

Box-Plots A Box plot is a graphical display that gives us information about the location of certain points in a set of data as well as the shape of the distribution of the data.

### Box-Plots :

Box-Plots The Upper Inner Fence is: UIF = Q3 + 1.5 (IQR) The Upper Outer Fence is: UOF = Q3 + 3.0 (IQR) The Lower Inner Fence is: LIF = Q1 - 1.5 (IQR) The Lower Outer Fence is: LOF = Q1 - 3.0 (IQR)

### Box-Plots :

Box-Plots The quartiles: Consider a data set rearranged in ascending order. The quartiles are those views( Q1, Q2, Q3) that divide the data set into four equal parts.

Quartiles

### Some useful formulas for calculating probabilities :

Some useful formulas for calculating probabilities Permutations Fundamental Counting Principle Combinations

Permutations

### Fundamental Counting Principle :

Fundamental Counting Principle

### Fundamental Counting Principle :

Fundamental Counting Principle

Combinations

Combinations

### 3-5 Discrete Uniform Distribution :

3-5 Discrete Uniform Distribution Definition

### 3-5 Discrete Uniform Distribution :

3-5 Discrete Uniform Distribution Example 3-13

### 3-5 Discrete Uniform Distribution :

3-5 Discrete Uniform Distribution Figure 3-7 Probability mass function for a discrete uniform random variable.

### 3-5 Discrete Uniform Distribution :

3-5 Discrete Uniform Distribution Mean and Variance

### 3-6 Binomial Distribution :

3-6 Binomial Distribution Random experiments and random variables

### 3-6 Binomial Distribution :

3-6 Binomial Distribution Random experiments and random variables

### 3-6 Binomial Distribution :

3-6 Binomial Distribution Definition

### 3-6 Binomial Distribution :

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

### 3-6 Binomial Distribution :

3-6 Binomial Distribution Example 3-18

### 3-6 Binomial Distribution :

3-6 Binomial Distribution Example 3-18

### 3-6 Binomial Distribution :

3-6 Binomial Distribution Definition

### 3-6 Binomial Distribution :

3-6 Binomial Distribution Example 3-19

### 3-7 Geometric and Negative Binomial Distributions :

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

### 3-7 Geometric and Negative Binomial Distributions :

3-7 Geometric and Negative Binomial Distributions Definition

### 3-7 Geometric and Negative Binomial Distributions :

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

### 3-7 Geometric and Negative Binomial Distributions :

3-7 Geometric and Negative Binomial Distributions Definition

### 3-7 Geometric and Negative Binomial Distributions :

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

### 3-7 Geometric and Negative Binomial Distributions :

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

### 3-7 Geometric and Negative Binomial Distributions :

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 :

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 Geometric and Negative Binomial Distributions 3-7.2 Negative Binomial Distribution

### 3-7 Geometric and Negative Binomial Distributions :

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

### 3-7 Geometric and Negative Binomial Distributions :

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

### 3-8 Hypergeometric Distribution :

3-8 Hypergeometric Distribution Definition

### 3-8 Hypergeometric Distribution :

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

### 3-8 Hypergeometric Distribution :

3-8 Hypergeometric Distribution Example 3-27

### 3-8 Hypergeometric Distribution :

3-8 Hypergeometric Distribution Example 3-27

### 3-8 Hypergeometric Distribution :

3-8 Hypergeometric Distribution Definition

### 3-8 Hypergeometric Distribution :

3-8 Hypergeometric Distribution Finite Population Correction Factor

### 3-8 Hypergeometric Distribution :

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

### 3-9 Poisson Distribution :

3-9 Poisson Distribution Example 3-30

### 3-9 Poisson Distribution :

3-9 Poisson Distribution Definition

### 3-9 Poisson Distribution :

3-9 Poisson Distribution Consistent Units

### 3-9 Poisson Distribution :

3-9 Poisson Distribution Example 3-33

### 3-9 Poisson Distribution :

3-9 Poisson Distribution Example 3-33

### 3-9 Poisson Distribution :

3-9 Poisson Distribution 