Discrete Random Variables & Distribution

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Published on November 3, 2008

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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 Slide 8: Example 3-5 Slide 9: Example 3-5 (continued) 3-3 Cumulative Distribution Functions : 3-3 Cumulative Distribution Functions Definition Slide 11: 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. Slide 16:  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) : 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 : Quartiles Some useful formulas for calculating probabilities : Some useful formulas for calculating probabilities Permutations Fundamental Counting Principle Combinations Permutations : Permutations Fundamental Counting Principle : Fundamental Counting Principle Fundamental Counting Principle : Fundamental Counting Principle Combinations : Combinations 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

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