AP Statistics Monday, 30 November 2015 OBJECTIVE TSW begin the study of discrete distributions. EVERYONE needs a calculator. The tests are graded.

Discrete Distributions

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

Two types of random variables: Discrete – count of some random variable (this unit) Continuous – measure of some random variable (1 st unit after Christmas Break)

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

Properties for a discrete probability distribution 1)For every possible x value, 0 < P(x) < 1. 2) For all values of x,  P(x) = 1.

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 X0123X0123 P(X)

Let x be the number of courses for which a randomly selected student at a certain university is registered. X P(X) ? P(x = 4) = P(x < 4) = What is the probability that the student is registered for at least five courses? P(x > 5) = 0.61

Formulas for mean & variance Found on formula card!

Let x be the number of courses for which a randomly selected student at a certain university is registered. X P(X) What is the mean and standard deviation of this distribution?  4.66  =

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! X0520 P(X)7/91/61/18 NO, since  = $1.944 which is less than it cost to play ($3).

AP Statistics Tuesday, 01 December 2015 OBJECTIVE TSW explore binomial distributions. TODAY’S ASSIGNMENTS –WS Binomial Distributions #1 Due tomorrow, Wednesday, 02 December –WS Binomial Distributions #2 Due on Friday, 04 December –WS Binomial Activity Due on Friday, 04 December Everyone needs a calculator.

AP Statistics Calendar for Remainder of Semester Mon, 11/30/15Discrete Distributions Tues, 12/01/15Binomial Distributions Wed, 12/02/15Geometric & Poisson Distributions Fri, 12/04/15QUIZ: Discrete & Binomial Distributions Mon, 12/07/15QUIZ: Geometric & Poisson Distributions Tues, 12/08/15Review: Discrete Distributions Wed, 12/09/15TEST: Discrete Distributions Fri, 12/ 11/15TEST: TEKS Exams Mon, 12/14/15Review – Final Tues, 12/15/15Review – Final Wed, 12/16/15Review – Final Fri, 12/18/15FINAL EXAM: 1 st Semester

Let x be the number of courses for which a randomly selected student at a certain university is registered. X P(X) What is the mean and standard deviation of this distribution?  4.66  =

Linear function of a random variable If x is a random variable and a and b are numerical constants, then the random variable y is defined by and The mean is changed by addition & multiplication! ONLY The standard deviation is ONLY changed by multiplication!

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?  = $  = $75.60

Linear combinations Just add or subtract the means! add 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 meanSD MC386 FR307 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  =

Special Discrete Distributions

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

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 have genes for O and A blood types; count the number of children with blood type O No, no fixed number

Toss a 3 coins and count the number of heads Find the discrete probability distribution X0123X0123 P(x) Out of 3 coins that are tossed, what is the probability of getting exactly 2 heads?

AP Statistics Wednesday, 02 December 2015 OBJECTIVE TSW explore binomial, geometric, and poisson distributions. Everyone needs a calculator. ASSIGNMENT DUE DATES –WS Binomial Distributions #1 Due by end of the period today. –WS Binomial Distributions #2 Due on Friday, 04 December –WS Binomial Activity Due on Friday, 04 December 2015.

Binomial Formula for Probability: Where:

Out of 3 coins that are tossed, what is the probability of getting exactly 2 heads? Formula substitution Probability statement

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

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)

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? Calculator strokes

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. P(X < 6) = binomcdf(10,0.3,6) =

Binomial formulas for mean and standard deviation

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?

ASSIGNMENTS WS Binomial WS #1 –Due by the end of the period today. WS Binomial WS #2 –Due Friday, 04 December Binomial Activity –Due on Friday, 04 December QUIZ: Discrete & Binomial Distributions on Friday, 04 December 2015.

Due on Friday, 04 December 2015.

In L1 – seq(x,x,0,10) In L2 – binompdf(10, 0.1,L1) Sketch histogram on board