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Understanding Special Distributions and Counting Rules in Statistics

This course section focuses on special probability distributions and the foundational counting methods crucial for calculating probabilities. Students will review key concepts from the last class and engage with chapter 5, covering probability trees, distribution functions, expected values, and standard deviations. The session will also introduce counting rules essential for determining possible outcomes in various scenarios, including binomial distributions and real-world examples. By the end, students will gain a solid understanding of how to apply these principles effectively.

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Understanding Special Distributions and Counting Rules in Statistics

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  1. Statistics & Data Analysis Course Number B01.1305 Course Section 31 Meeting Time Wednesday 6-8:50 pm CLASS #4

  2. Class #4 Outline • Brief review of last class • Questions on homework • Chapter 5 – Special Distributions

  3. Review of Last Class • Probability trees • Probability distribution functions • Expected value • Standard deviation

  4. Chapter 5 Some Special Probability Distributions

  5. Chapter Goals • Introduce some special, often used distributions • Understand methods for counting the number of sequences • Understand situations consisting of a specified number of distinct success/failure trials • Understanding random variables that follow a bell-shaped distribution

  6. Counting Possible Outcomes • In order to calculate probabilities, we often need to count how many different ways there are to do some activity • For example, how many different outcomes are there from tossing a coin three times? • To help us to count accurately, we need to learn some counting rules • Multiplication Rule : If there are m ways of doing one thing and n ways of doing another thing, there are m times n ways of doing both

  7. Example • An auto dealer wants to advertise that for $20G you can buy either a convertible or 4-door car with your choice of either wire or solid wheel covers. • How many different arrangements of models and wheel covers can the dealer offer?

  8. Counting Rules • Recall the classical interpretation of probability:P(event) = number of outcomes favoring event / total number of outcomes • Need methods for counting possible outcomes without the labor of listing entire sample space • Counting methods arise as answers to: • How many sequences of k symbols can be formed from a set of r distinct symbols using each symbol no more than once? • How many subsets of k symbols can be formed from a set of r distinct symbols using each symbol no more than once? • Difference between a sequence and a subset is that order matters for a sequence, but not for a subset

  9. Counting Rules (cont) • Create all k=3 letter subsets and sequences of the r=5 letters: A, B, C, D and E • How many sequences are there? • How many subsets are there?

  10. Counting Rules (cont)

  11. Review: Sequence and Subset • For a sequence, the order of the objects for each possible outcome is different • For a subset, order of the objects is not important

  12. Example • A group of three electronic parts is to be assembled into a plug-in unit for a TV set • The parts can be assembled in any order • How many different ways can they be assembled? • There are eight machines but only three spaces on the machine shop floor. • How many different ways can eight machines be arranged in the three available spaces? • The paint department needs to assign color codes for 42 different parts. Three colors are to be used for each part. How many colors, taken three at a time would be adequate to color-code the 42 parts?

  13. Binomial Distribution • Percentages play a major role in business • When percentage is determined by counting the number of times something happens out of the total possibilities, the occurrences might following a binomial distribution • Examples: • Number of defective products out of 10 items • Of 100 people interviewed, number who expressed intention to buy • Number of female employees in a group of 75 people • Of all the stocks trades on the NYSE, the number that went up yesterday

  14. Binomial Distribution (cont) • Each time the random experiment is run, either the event happens or it doesn’t • The random variable X, defined as the number of occurrences of a particular event out of n trials has a binomial distribution if: • For each of the n trials, the event always has the same probability  of happening • The trials are independent of one another

  15. Example: Binomial Distribution • You are interested in the next n=3 calls to a catalog order desk and know from experience that 60% of calls will result in an order • What can we say about the number of calls that will result in an order? • Questions: • Create a probability tree • Create a probability distribution table • What is the expected number of calls resulting in an order? • What is the standard deviation?

  16. Binomial Distribution the Easy Way

  17. Finding Binomial Probabilities

  18. Example: Binomial Probabilities • How many of your n=6 major customers will call tomorrow? • There is a 25% chance that each will call • Questions: • How many do you expect to call? • What is the standard deviation? • What is the probability that exactly 2 call? • What is the probability that more than 4 call?

  19. Example • It’s been a terrible day for the capital markets with losers beating winners 4 to 1 • You are evaluating a mutual fund comprised of 15 randomly selected stocks and will assume a binomial distribution for the number of securities that lost value • Questions: • What assumptions are being made? • What is the random variable? • How many securities do you expect to lose value? • What is the standard deviation of the random variable? • Find the probability that 8 securities lose value • What is the probability that 12 or more lose value?

  20. The Normal Distribution

  21. Normal Distribution • The normal distribution is sometimes called a Gaussian Distribution, after its inventor, C. F. Gauss (1777- 1855). • Well-known “bell-shaped” distribution • Mean and standard deviation determine center and spread of the distribution curve • The mathematical formula for the normal f (y) is given in HO, p. 157. We won't be needing this formula; just tables of areas under the curve. • The empirical rule holds for all normal distributions • Probability of an event corresponds to area under the distribution curve

  22. Standard Normal Distribution • Normal Distribution with =0 and =1 • Letter Z is used to denote a random variable that follows a Standard Normal Distribution

  23. Visualization Symmetrical Tail Tail Mean, Median and Mode

  24. Characteristics • Bell-shaped with a single peak at the exact center of the distribution • Mean, median and mode are equal and located at the peak • Symmetrical about the mean • Falls off smoothly in both directions, but the curve never actually touches the X-axis

  25. Why Its Important • Many psychological and educational variables are distributed approximately normally • Measures of reading ability, introversion, job satisfaction, and memory are among the many psychological variables approximately normally distributed • Although the distributions are only approximately normal, they are usually quite close. • It is easy for mathematical statisticians to work with • This means that many kinds of statistical tests can be derived for normal distributions • Almost all statistical tests discussed in this text assume normal distributions • These tests work very well even if the distribution is only approximately normally distributed.

  26. More Visualizations =3.1 years, Plant A =3.9 years, Plant B =5 years, Plant C

  27. Z-score • Compute probabilities using tables or computer • Convert to z-score: • Look up CUMULATIVE PROBABILITY ON TABLE:

  28. Determining Probabilities

  29. LOOKUP Table Standard Normal Lookup Table

  30. Example • Sales forecasts are assumed to follow a normal distribution • Target, or expected value is $20M with a $3M standard deviation • What is the probability of sales lower than $15M? • What is the probability sales exceed $25M? • What is the probability sales are between $15M and $25M ?? • What is the value of k such that the sales forecast exceeds k is 60% ?

  31. Example • Benefits compensation costs for employees with a certain financial services firm are approximately normally distributed with a mean of $18,600 and standard deviation of $2,700. • Find the probability that an employee chosen at random has an benefits package that costs less than $15,000 • Find the probability that an employee chosen at random has an benefits package that costs more than $21,000 • What is the value of k such that the benefits compensation exceeds k is 95% ?

  32. Example • A telephone-sales firm is considering purchasing a machine that randomly selects and automatically dials telephone numbers • The firm would be using the machine to call residences during the evening; calls to business phones would be wasted. • The manufacturer of the machine claims that its programming reduces the business-phone rate to 15% • As a test, 100 phone numbers are to be selected at random from a very large set of possible numbers • Are the binomial assumptions satisfied? • Find the probability that at least 24 of the numbers belong to business phones • If in fact 24 of the 100 numbers turn out to be business phones, does that cast series doubt on the manufacturer’s claim? • Find the expected value and standard deviation of the number of business phone numbers in the sample

  33. Example • Assumed the stock market closed at 8,000 yesterday. • Today you expect the market to rise a mean of 1 point, with a standard deviation of 34 points. Assume a normal distribution. • What is the probability the market goes down tomorrow? • What is the probability the market goes up more than 10 points tomorrow? • What is the probability the market goes up more than 40 points tomorrow? • What is the probability the market goes up more than 60 points tomorrow? • Find the probability that the market changes by more than 20 points in either direction. • What is the value of k such that the market close exceeds k is 75% ?

  34. Using R • factorial(n) – n! • dbinom(x, n, p) – binomial probability distribution function • pbinom(x, n, p) – binomial cumulative distribution function • pnorm(q, mean, sd) – normal cumulative distribution function • qnorm(p, mean, sd) – inverse CDF

  35. Homework #4 Hildebrand/Ott 5.2, page 141 5.3, page 141 5.9, page 150 5.14, page 150 5.32, page 163 5.33, page 163 5.34, page 163 Reading: Chapter 6 (all) and 7 (all). Verzani 6.5

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