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Chapter 7

Chapter 7. Discrete Distributions. Random Variable -. A numerical variable whose value depends on the outcome of a chance experiment. Two types:. Discrete – count of some random variable Continuous – measure of some random variable. Discrete Probability Distribution.

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Chapter 7

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  1. Chapter 7 Discrete Distributions

  2. Random Variable - • A numerical variable whose value depends on the outcome of a chance experiment

  3. Two types: • Discrete – count of some random variable • Continuous – measure of some random variable

  4. Discrete Probability Distribution • Gives the probabilities associated with each possible x value • Usually displayed in a table, but can be displayed with a histogram or formula

  5. Discrete probability distributions 3)For every possible x value, 0 < P(x) < 1. 4) For all values of x, S P(x) = 1.

  6. Suppose you toss 3 coins & record the number of heads. The random variable X defined as ... Create a probability distribution. Create a probability histogram. The number of heads tossed X 0 1 2 3 P(X) .125 .375 .375 .125

  7. Let x be the number of courses for which a randomly selected student at a certain university is registered. X 1 2 3 4 5 6 7 P(X) .02 .03 .09 ? .40 .16 .05 P(x = 4) = P(x < 4) = P(x < 4) = What is the probability that the student is registered for at least five courses? Why does this not start at zero? .25 .14 P(x > 5) = .61 .39

  8. Formulas for mean & variance Found on formula card!

  9. Let x be the number of courses for which a randomly selected student at a certain university is registered. X 1 2 3 4 5 6 7 P(X) .02 .03 .09 .25 .40 .16 .05 What is the mean and standard deviations of this distribution? m = 4.66 & s = 1.2018

  10. X 0 5 20 P(X) 7/9 1/6 1/18 Here’s a game: If a player rolls two dice and gets a sum of 2 or 12, he wins $20. If he gets a 7, he wins $5. The cost to roll the dice one time is $3. Is this game fair? A fair game is one where the cost to play EQUALS the expected value! NO, since m = $1.944 which is less than it cost to play ($3).

  11. If x is a random variable, then the random variable y is defined by and Linear transformation of a random variable The mean is changed by addition & multiplication! The standard deviation is ONLY changed by multiplication!

  12. Let x be the number of gallons required to fill a propane tank. Suppose that the mean and standard deviation is 318 gal. and 42 gal., respectively. The company is considering the pricing model of a service charge of $50 plus $1.80 per gallon. Let y be the random variable of the amount billed. What is the mean and standard deviation for the amount billed? m = $622.40 & s = $75.60

  13. Linear combinations Just add or subtract the means! If independent, always add the variances!

  14. A nationwide standardized exam consists of a multiple choice section and a free response section. For each section, the mean and standard deviation are reported to be mean SD MC 38 6 FR 30 7 If the test score is computed by adding the multiple choice and free response, then what is the mean and standard deviation of the test? m = 68 & s = 9.2195

  15. Special Discrete Distributions

  16. Binomial Distribution B(n,p) • Each trial results in one of two mutually exclusive outcomes. (success/failure) • There are a fixed number of trials • Outcomes of different trials are independent • The probability that a trial results in success is the same for all trials The binomial random variable x is defined as the number of successes out of the fixed number

  17. Are these binomial distributions? • Toss a coin 10 times and count the number of heads Yes • Deal 10 cards from a shuffled deck and count the number of red cards No, probability does not remain constant • Two parents with genes for O and A blood types and count the number of children with blood type O No, no fixed number

  18. Binomial Formula: Where:

  19. Out of 3 coins that are tossed, what is the probability of getting exactly 2 heads?

  20. The number of inaccurate gauges in a group of four is a binomial random variable. If the probability of a defect gauge is 0.1, what is the probability that only 1 is defective? More than 1 is defective?

  21. Binomialpdf(n,p,x) – this calculates the probability of a single binomial P(x = k) Binomialcdf(n,p,x) – this calculates the cumulative probabilities from P(0) to P(k) OR P(X < k) Calculator

  22. A genetic trait of one family manifests itself in 25% of the offspring. If eight offspring are randomly selected, find the probability that the trait will appear in exactly three of them. At least 5?

  23. In a certain county, 30% of the voters are Republicans. If ten voters are selected at random, find the probability that no more than six of them will be Republicans. B(10,.3) P(x < 6) = binomcdf(10,.3,6) = .9894 What is the probability that at least 7 are not Republicans? P(x > 7) = 1 - binomcdf(10,.7,6) = .6496

  24. Binomial formulas for mean and standard deviation

  25. In a certain county, 30% of the voters are Republicans. How many Republicans would you expect in ten randomly selected voters? What is the standard deviation for this distribution? expect

  26. In a certain county, 30% of the voters are Republicans. What is the probability that the number of Republicans out of 10 is within 1 standard deviation of the mean? B(10,.3) P(1.55 < x < 4.45) = binomcdf(10,.3,4) – binomcdf(10,.3,1) = .7004

  27. X Geometric Distributions: • There are two mutually exclusive outcomes • Each trial is independent of the others • The probability of success remains constant for each trial. The random variable x is the number of trials UNTIL the FIRST success occurs. So what are the possible values of X How far will this go? To infinity . . . 1 2 3 4

  28. Differences between binomial & geometric distributions • The difference between binomial and geometric properties is that there is NOT a fixed number of trials in geometric distributions!

  29. Other differences: • Binomial random variables start with 0 while geometric random variables start with 1 • Binomial distributions are finite, while geometric distributions are infinite

  30. Geometric Formulas: Not on formula sheet – they will be given on quiz or test

  31. Count the number of boys in a family of four children. What are the values for these random variables? Binomial: X 0 1 2 3 4 Count children until first son is born Geometric: X 1 2 3 4 . . .

  32. Calculator • geometpdf(p,x) – finds the geometric probability for P(X = k) • Geometcdf(p,x) – finds the cumulative probability for P(X < k) No “n” because there is no fixed number!

  33. What is the probability that the first son is the fourth child born? What is the probability that the first son born is at most the fourth child? What is the probability that the first son born is at least the third child?

  34. A real estate agent shows a house to prospective buyers. The probability that the house will be sold to the person is 35%. What is the probability that the agent will sell the house to the third person she shows it to? How many prospective buyers does she expect to show the house to before someone buys the house? SD?

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