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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 1) Gives the probabilities associated with each possible x value 2) Usually displayed in a table, but can be displayed with a histogram or formula Discrete probability distributions 3)For every possible x value, 0 < P(x) < 1. 4) For all values of x, S P(x) = 1. Suppose you toss 3 coins & record the number of heads. The random variable X defined as ... The number of heads tossed Create a probability distribution. X P(X) 0 .125 1 .375 2 .375 3 .125 Create a probability histogram. Let x be the number of courses for which a randomly selected student at a certain university is registered. X 1 2 3 P(X).02 .03 .09 4 5 6 7 ? .40 .16 .05 Why does this not start at zero? .25 P(x = 4) = P(x < 4) = .14 P(x < 4) = .39 P(x > 5) = .61 What is the probability that the student is registered for at least five courses? Formulas for mean & variance x x x 2 xi p i i x p 2 i Found on formula card! 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? 4.66 & = 1.2018 Here’s a game: If a player rolls two dice and gets aAsum of 2is or 12, fair game one where the cost to play EQUALS he wins $20.theIf he gets a expected value! 0 $5. 5The cost 20 7,X he wins to P(X) 7/9 1/6 1/18 roll the dice one time is $3. Is this game fair? NO, since = $1.944 which is less than it cost to play ($3). Linear transformation of a random variable The mean is changed If x is a random variable, then the by addition & random variable y is defined by multiplication! y a bx The standard deviation is and ONLY changed y a bx by multiplication! b y x 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? = $622.40 & = $75.60 JustLinear add or subtract combinations the means! a b a b a b a b 2 a a b 2 b If independent, always add the variances! 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? 68 & = 9.2195 Special Discrete Distributions Binomial Distribution B(n,p) 1. Each trial results in one of two mutually exclusive outcomes. (success/failure) 2. There are a fixed number of trials 3. Outcomes of different trials are independent 4. 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 Are these binomial distributions? 1) Toss a coin 10 times and count the number of heads Yes 2) Deal 10 cards from a shuffled deck and count the number of red cards No, probability does not remain constant 3) Two parents with genes for O and A blood types and count the number of children with blood type O No, no fixed number Binomial Formula: n k n k P (x k ) p 1 p k Where: n n C k k Out of 3 coins that are tossed, what is the probability of getting exactly 2 heads? B (3,.5) 3 2 1 P (x 2) 0.5 0.5 .375 2 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? B ( 4,.1) 4 P (x 1) 0.11 0.93 .2916 1 More than 1 is defective? P (x 1) 1 (P (0) P (1)) .0523 Calculator • 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) 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. B (8,.25) P (X 3) binompdf (8,.25,3) .2076 At least 5? P (X 5) 1 binomcdf (8,.25,4) .0273 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 Binomial formulas for mean and standard deviation x np x np 1 p 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? B (10,.3) x 10(.3) 3 Republicans x 10(.3)(.7) 1.45 Republicans 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 Geometric Distributions: 1. There are two mutually So what are the exclusive outcomes How far this go? To will infinity possible values of X 2. Each trial is independent of the others 3. The probability of success remains constant for each trial. X random . 1 2 variable 3 4 x.is. the The number of trials UNTIL the FIRST success occurs. 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! Other differences: •Binomial random variables start with 0 while geometric random variables start with 1 •Binomial distributions are finite, while geometric distributions are infinite Geometric Formulas: P (x ) p 1 p x 1 1 x p 1p x 2 p Not on formula sheet – they will be given on quiz or test Count the number of boys in a family of four children. Binomial: X 0 1 2 3 What are the values for these random 4 variables? Count children until first son is born Geometric: X 1 2 3 4 . . . Calculator • geometpdf(p,x) – finds the geometric probability for P(X = k) No “n” because there is no • Geometcdf(p,x) – finds the fixed number! cumulative probability for P(X < k) What is the probability that the first son is the fourth child born? P (X 4) geometricpdf (.5,4) .0625 What is the probability that the first son born is at most the fourth child? P (X 4) geometriccdf (.5,4) .9375 What is the probability that the first son born is at least the third child? P (X 3) 1 geometriccdf (.5,2) .25 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? P (x 3) geometricpdf (.35,3) .1479 How many prospective buyers does she expect to show the house to before someone buys the house? SD? 1 1 .35 x 2.86 buyers x 2.304 buyers 2 .35 .35