1 / 28

QUANTITATIVE ANALYSIS

QUANTITATIVE ANALYSIS. UNIT 8: Discrete Probability Distributions. TouchText. Sample Spaces and Random Variables Probability Distributions of Random Variables Mean, Variance and Standard Deviation of Random Variables The Binomial Distribution. Problems and Exercises. Next.

israel
Télécharger la présentation

QUANTITATIVE ANALYSIS

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. QUANTITATIVE ANALYSIS UNIT 8: Discrete Probability Distributions TouchText • Sample Spaces and Random Variables • Probability Distributions of Random Variables • Mean, Variance and Standard Deviation of Random Variables • The Binomial Distribution Problems and Exercises Next

  2. Random Experiments, Events and Sample Spaces Random Experiments are those whose outcomes are determined by chance (luck). Outcomes of Random Experiments are also called “Events”. All possible events taken together constitute the experiment’s Sample Space. Dictionary Outcome 1 = Event 1 Outcome 2 = Event 2 Outcome 3 = Event 3 Outcome 4 = Event 4 Etc. Random Experiment Sample Space Event = what does, in fact, happen. Sample Space = everything that might happen. Back Next

  3. Sample Spaces and Random Variables Using numeric random variables to represent outcomes in a sample space is very useful for statistical analysis. A Random Variable assigns one (only) numeric value to each outcome in the sample space. Dictionary Random Variable “Z” z = 0 z = 1 z = 2 Random Experiment Event Event Event Event *Big “Z” for random variable; small “z” for value of random variable. Sample Space Each event gets assigned only one random variable value. However, a random variable value can accept more than one event. Back Next

  4. Sample Spaces and Random Variables: Example: Coin Toss Example: Suppose a coin is tossed three times: Random Variable “Z” = number of “heads” out of 3 coin tosses Dictionary TTT TTH THT HTT THH HTH HHT HHH z = 0 z = 1 z = 2 z = 3 Coin Toss (3 times) Sample Space Each event gets assigned only one random variable value. However, a random variable value can accept more than one event. Back Next

  5. Sample Spaces and Random Variables: Example 2: Coin Toss Example: Suppose a coin is tossed three times: Random Variable “X”: X = 1 if all tosses the same; X = 0 otherwise Dictionary TTT TTH THT HTT THH HTH HHT HHH x = 0 x = 1 Coin Toss (3 times) Sample Space Each event gets assigned only one random variable value. However, a random variable value can accept more than one event. Back Next

  6. Sample Spaces and Random Variables: Example 3: Turning on the TV Example: When you turn on the TV, either the TV will open with a scheduled program in progress, or with a commercial advertisement. Dictionary Sample Space Random Variable “Z”: z = 0 if commercial; z = 1 if program Commercial Scheduled Program z = 0 z = 1 *For future analysis, assume that commercials take up exactly 20% of total TV time. Back Next

  7. Sample Spaces and Random Variables: Example 4: Roll of a Pair of Dice Example: Suppose a pair of (2) dice is rolled: Random Variable “X”: X = 1 if both dice are the same; X = 0 otherwise Dictionary Dice #1Dice #2 11, 2, 3, 4, 5, 6 2 1, 2, 3, 4, 5, 6 31, 2, 3, 4, 5, 6 41, 2, 3, 4, 5, 6 51, 2, 3, 4, 5, 6 61, 2, 3, 4, 5, 6 30white/black pairs x = 0 x = 1 6white/whitepairs Sample Space Each event gets assigned only one random variable value. However, a random variable value can accept more than one event. Back Next

  8. Random Variables and Probabilities Random events each have a probability, to which probability theory applies. Dictionary Random Variable Sample Space Values of Random Variable Random Events Random Experiment Random Variable Probabilities Random variables “inherit” their probabilities from the random events to which they are assigned. Event Probabilities Probability Theory Probability Theory Back Next

  9. Sample Spaces and Random Variables: Example: Coin Toss Example: Suppose a coin is tossed three times: Random Variable “Z” = number of “heads” out of 3 coin tosses Dictionary Pr.(TTT) = 1/8 Pr.(TTH) = 1/8 Pr.(THT) = 1/8 Pr.(HTT) = 1/8 Pr.(THH) = 1/8 Pr.(HTH) = 1/8 Pr.(HHT) = 1/8 Pr.(HHH) = 1/8 Pr.(z=0) = 1/8 Pr.(z=1) = 3/8 Pr.(z=2) = 3/8 Pr.(z=3) = 1/8 Total = 8/8 Coin Toss (3 times) Sample Space Back Next

  10. Sample Spaces and Random Variables: Example: Coin Toss Example: Suppose a coin is tossed three times: Random Variable “X”: X = 1 if all tosses the same; X = 0 otherwise Dictionary Pr.(TTT) = 1/8 Pr.(TTH) = 1/8 Pr.(THT) = 1/8 Pr.(HTT) = 1/8 Pr.(THH) = 1/8 Pr.(HTH) = 1/8 Pr.(HHT) = 1/8 Pr.(HHH) = 1/8 Pr(x=0) = 6/8 Pr(x=1) = 2/8 Total = 8/8 Coin Toss (3 times) Sample Space Each event gets assigned only one random variable value. However, a random variable value can accept more than one event. Back Next

  11. Sample Spaces and Random Variables: Example 4: Roll of a Pair of Dice Example: Suppose a pair of (2) dice is rolled: Random Variable “X”: X = 1 if both dice are the same; X = 0 otherwise Dictionary Dice #1Dice #2 11, 2, 3, 4, 5, 6 2 1, 2, 3, 4, 5, 6 31, 2, 3, 4, 5, 6 41, 2, 3, 4, 5, 6 51, 2, 3, 4, 5, 6 61, 2, 3, 4, 5, 6 30white/black pairs Pr(x=0)=30/36 Pr(x=1) = 6/36 Total = 36/36 6white/whitepairs Sample Space There are a total of 36 possible outcomes, each with 1/36 probability. Back Next

  12. Mutually Exclusive Outcomes Because (a) each event is mutually exclusive, and (b) each event gets assigned to only one random variable value, then… The values of the random variables are mutually exclusive. Dictionary Back Next

  13. Discrete Frequency Distributions of Random Variables: Example 1 Because random variables are quantitative (numeric) and have well defined probabilities, these probabilities can be characterized in a discrete (relative) frequency distribution. Pr.(z=0) = 1/8 Pr.(z=1) = 3/8 Pr.(z=2) = 3/8 Pr.(z=3) = 1/8 Total = 8/8 Dictionary Random Variable “Z” = number of “heads” out of 3 coin tosses Back Next

  14. Discrete Frequency Distributions of Random Variables: Example 2 Example: Suppose a pair of (2) dice is rolled: Pr(x=0)=30/36 Pr(x=1) = 6/36 Total = 36/36 Random Variable “X”: X = 1 if both dice are the same; X = 0 otherwise 30white/black pairs Dictionary 6white/whitepairs Back Next

  15. Descriptive Statistics: The Mean Recall the beginning of this course: Dictionary Unit 1: Visually Presenting Data Unit 2: Descriptive Statistics: Central Tendency Unit 3: Descriptive Statistics: Dispersion We’ve just presented the data random variables as probability (relative frequency) distributions. Next we need to describe the data in terms of central tendency (mean) and dispersion (variance or standard deviation). Back Next

  16. The Mean Value of a Random Variable Example 1 The mean value (or average) of a random variable is calculated as: Mean(X) = μX= E[X] = Dictionary different notation Example: Random Variable “Z” = number of “heads” out of 3 coin tosses. Pr.(z=0) = 1/8 Pr.(z=1) = 3/8 Pr.(z=2) = 3/8 Pr.(z=3) = 1/8 Total = 8/8 So out of 3 coin tosses, the average number of heads is 1.5. Back Next

  17. The Mean Value of a Random Variable Example 2 Pr(x=0)=30/36 Pr(x=1) = 6/36 Total = 36/36 Random Variable “X”: X = 1 if both dice are the same; X = 0 otherwise 30white/black pairs Dictionary 6white/whitepairs Mean(X) = μX= E[X] = So, a person should be roll a double about one out of six times. Back Next

  18. The Variance and Standard Deviation of a Random Variable One measure of the dispersion, or spread, of a distribution is its Variance. Dictionary Perhaps more useful that the variance is the Standard Deviation, which is simply the square root of the variance. * If you’ll recall, the standard deviation is often preferred because it is measured in the same units as the original data. * With respect to previous formulas, these are the population variance and standard deviation formulas. Back Next

  19. The Variance and Standard Deviation of a Random Variable: Example 1 Example: Random Variable “Z” = number of “heads” out of 3 coin tosses. Dictionary Pr.(z=0) = 1/8 = 0.125 Pr.(z=1) = 3/8 = 0.375 Pr.(z=2) = 3/8 = 0.375 Pr.(z=3) = 1/8 = 0.125 Total = 8/8 Also: Back Next

  20. The Variance and Standard Deviation of a Random Variable: Example 2 Pr(x=0)=30/36 Pr(x=1) = 6/36 Total = 36/36 Random Variable “X”: X = 1 if both dice are the same; X = 0 otherwise Dictionary And so Back Next

  21. A Bernoulli Experiment A Bernoulli Experiment is a random experiment whose sample space is only 2 simple events. Dictionary Heads/tails True/false Yes/no A Bernoulli Trial is one of a series of Bernoulli Experiments. However, it is required that the various experiments are independent, so that the probabilities of the two outcomes remain constant from one experiment to the next. In turn, a Binomial Random Variable represents the number of “successes” from the various Bernoulli Trials. Back Next

  22. The Binomial Experiment A Binomial Experiment consists of … Dictionary • Two possible outcomes, “success” and “failure” • A fixed number “n” of trials • The probability of success, “p”, stays the same from trial to trial • The trials are independent • The Binomial Random Variable is the number of successes (p) in the (n) trials. * The terms “success” and “failure” do not necessarily mean “good” and “bad”. Rather, think of “success” as the outcome of interest – the outcome you wish to measure. Back Next

  23. Generating Binomial Probabilities The function that generates binomial probabilities is Dictionary • Where • n = number of trials • p = probability of success • x = observed success in n trials Mathematically, the exclamation point “!” means factorial. 1! = 1; 2! = 2x1 = 2; 3! = 3x2x1 = 6 4! = 4x3x2x1 = 24 etc. Back Next

  24. Generating Binomial Probabilities: Example Example: A student takes a 20 question, multiple choice exam. There are 4 possible answers to each question, and the student guesses at each one. (Thus, he has a 25% chance of getting each question right.) What is the probability that the student gets exactly 3 question out of 20 correct? Dictionary Back Next

  25. Binomial Calculators There are too many possible combinations of x, p, and n to produce a table. However, Binomial calculators are available on the internet. Helpfully, these can calculate (a) the probability of exactly x successes, (b) the probability of x or less successes; and (c) the probability of at least x successes. Binomial Helpful Binomial Calculator: http://www.vassarstats.net/textbook/ch5apx.html Back Next

  26. The Mean, Variance and Standard Deviation of a Binomial Distribution Fortunately, the formulas for the mean, variance and standard deviation of a binomial distribution are relatively simple. Dictionary Binomial Statistics: Mean μ = np Variance σ2= np(1-p) Standard Deviation σ= Back Next

  27. QUANTITATIVE ANALYSIS Dictionary Back Next

  28. End of Unit 8 Questions and Problems The following problems require the calculation of various statistics using MS Excel. The problems are linked to actual Excel spreadsheets, where students should do their work. Dictionary Take Notes Back End

More Related