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

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3-1 Discrete Random Variables

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  1. 3-1 Discrete Random Variables

  2. 3-1 Discrete Random Variables Example 3-1

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

  4. 3-2 Probability Distributions and Probability Mass Functions Figure 3-2Loadings at discrete points on a long, thin beam.

  5. 3-2 Probability Distributions and Probability Mass Functions Definition

  6. Example 3-5

  7. Example 3-5 (continued)

  8. 3-3 Cumulative Distribution Functions Definition

  9. Example 3-8

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

  11. 3-4 Mean and Variance of a Discrete Random Variable Definition

  12. 3-4 Mean and Variance of a Discrete Random Variable Figure 3-5A 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.

  13. 3-4 Mean and Variance of a Discrete Random Variable Figure 3-6The probability distribution illustrated in Parts (a) and (b) differ even though they have equal means and equal variances.

  14. Example 3-11

  15. 3-4 Mean and Variance of a Discrete Random Variable Expected Value of a Function of a Discrete Random Variable

  16. 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

  17. 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.

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

  19. 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.

  20. 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.

  21. 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.

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

  23. 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

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

  25. 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.

  26. 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)

  27. 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.

  28. Quartiles

  29. Some useful formulas for calculating probabilities • Permutations • Fundamental Counting Principle • Combinations

  30. Permutations

  31. Fundamental Counting Principle

  32. Fundamental Counting Principle

  33. Combinations

  34. Combinations

  35. 3-5 Discrete Uniform Distribution Definition

  36. 3-5 Discrete Uniform Distribution Example 3-13

  37. 3-5 Discrete Uniform Distribution Figure 3-7Probability mass function for a discrete uniform random variable.

  38. 3-5 Discrete Uniform Distribution Mean and Variance

  39. 3-6 Binomial Distribution Random experiments and random variables

  40. 3-6 Binomial Distribution Random experiments and random variables

  41. 3-6 Binomial Distribution Definition

  42. 3-6 Binomial Distribution Figure 3-8Binomial distributions for selected values of n and p.

  43. 3-6 Binomial Distribution Example 3-18

  44. 3-6 Binomial Distribution Example 3-18

  45. 3-6 Binomial Distribution Definition

  46. 3-6 Binomial Distribution Example 3-19

  47. 3-7 Geometric and Negative Binomial Distributions Example 3-20

  48. 3-7 Geometric and Negative Binomial Distributions Definition

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