AP StatisticsMonday, 30 November 2015

• OBJECTIVE TSW begin the study of discrete distributions.

• EVERYONE needs a calculator.

• The tests are graded.

Discrete Distributio

ns

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 (1st 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, S 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

X 0 1 2 3P(X) 0.125 0.375 0.375 0.125

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) 0.02 0.03 0.09 ? 0.40 0.16 0.05

P(x = 4) = P(x < 4) =P(x < 4) =What is the probability that the student is registered for at least five courses?

0.250.140.39

P(x > 5) = 0.61

Formulas for mean & variance

x i ix p

Found on formula card!

22x i x i

x p

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) 0.02 0.03 0.09 0.25 0.40 0.16 0.05

What is the mean and standard deviation of this distribution?

4.66 = 1.2018

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!X 0 5 20P(X) 7/9 1/61/18

NO, since = $1.944 which is less than it cost to play ($3).

AP StatisticsTuesday, 01 December 2015

• OBJECTIVE TSW explore binomial distributions.

• TODAY’S ASSIGNMENTS– WS Binomial Distributions #1

• Due tomorrow, Wednesday, 02 December 2015.– WS Binomial Distributions #2

• Due on Friday, 04 December 2015.– WS Binomial Activity

• Due on Friday, 04 December 2015.

• Everyone needs a calculator.

AP StatisticsCalendar for Remainder of Semester

• Mon, 11/30/15 Discrete Distributions• Tues, 12/01/15 Binomial Distributions• Wed, 12/02/15 Geometric & Poisson Distributions• Fri, 12/04/15 QUIZ: Discrete & Binomial Distributions

• Mon, 12/07/15 QUIZ: Geometric & Poisson Distributions

• Tues, 12/08/15 Review: Discrete Distributions• Wed, 12/09/15 TEST: Discrete Distributions• Fri, 12/ 11/15TEST: TEKS Exams

• Mon, 12/14/15 Review – Final• Tues, 12/15/15 Review – Final• Wed, 12/16/15 Review – Final• Fri, 12/18/15 FINAL EXAM: 1st Semester

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) 0.02 0.03 0.09 0.25 0.40 0.16 0.05

What is the mean and standard deviation of this distribution?

4.66 = 1.2018

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

• andbxay

xyxbxay

xbxay

bb

ba

or2222

The mean is changed by addition

& multiplication!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? = $622.40

= $75.60

Linear combinations

1 2I f ... thenny x x x

Just add or subtract the means!

If independent, always add the variances!

y 1 2 ...x x xn

2 2 21 2 ...y x x xn

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 SDMC 38 6FR 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)• 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 headsYes

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 distributionX 0 1 2 3P(x) 0.125 0.375 0.375 0.125

Out of 3 coins that are tossed, what is the probability of getting exactly 2 heads? P X=2 =0.375

AP StatisticsWednesday, 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 2015.

– WS Binomial Activity• Due on Friday, 04 December 2015.

Binomial Formula for Probability:

( ) 1 n kknP x k p pk

Where:

Cn knk

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

2 13 2P(X 2) C 0.5 0.5 0.375

Formulasubstitution

Probabilitystatement

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?

1 34 1P X 1 C 0.1 0.9 0.2916

P X 1 1 P 0 P 1 0.0523

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)

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?

P X 3 binomialpdf 8,0.25,3 0.2076

P X 5 1 binomialcdf 8,0.25,4 0.0273 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) = 0.9894

1x np p

Binomial formulas for mean and standard

deviation

x np

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?

10 0.3 3 Republicansx

10 0.3 0.7 1.45 Republicansx

ASSIGNMENTS• WS Binomial WS #1

– Due by the end of the period today.

• WS Binomial WS #2– Due Friday, 04 December 2015.

• Binomial Activity– Due on Friday, 04 December 2015.

• 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