discrete random variables chapter 4 objectives 1

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CHAPTER 4 Discrete Random Variables Chapter 4 Objectives 1

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Page 1: Discrete Random Variables Chapter 4 Objectives 1

CHAPTER 4Discrete Random Variables

Chapter 4 Objectives

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Page 2: Discrete Random Variables Chapter 4 Objectives 1

CHAPTER 4 OBJECTIVES

The student will be able to Recognize and understand

discrete probability distribution functions, in general.

Recognize the Binomial probability distribution and apply it appropriately.

Calculate and interpret expected value (average).

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Page 3: Discrete Random Variables Chapter 4 Objectives 1

DISCRETE RANDOM VARIABLES

TypesGeneralBinomialPoisson (not doing)Geometric (not doing)Hypergeometric (not doing)

Calculator becomes major tool

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Page 4: Discrete Random Variables Chapter 4 Objectives 1

GENERAL DISCRETE VARIABLES

Probability Distribution Function (PDF)Characteristics

each probability is between 0 and 1, inclusive

the sum of the probabilities is 1An edit of the Relative

Frequency Table where the Rel Freq column is relabeled P(X) and we drop the Freq and Cum Freq columns

Calculated from the PDF Mean (expected value) Standard Deviation

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An example

Page 5: Discrete Random Variables Chapter 4 Objectives 1

BINOMIAL Characteristics

each probability is between 0 and 1, inclusive

the sum of the probabilities is 1fixed number of trialsonly 2 possible outcomes for each trial, probabilities, p

and q, remain the same (p + q = 1)

Other factsX ~ B(n, p)X = number of successesn = number of independent

trialsx = 0,1,2,…,nµ = npσ = sqrt(npq)

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Problem 8

Page 6: Discrete Random Variables Chapter 4 Objectives 1

USING CALCULATOR FOR BINOMIAL

What the calculator can do Find P(X = x)

Binompdf(n, p, x) Find P(X < x)

Binomcdf(n, p, x)

What the calculator needs help with Find P(X < x) = P(X < x-1)

Binomcdf(n, p, x-1) Find P(X > x) = 1 – P(X < x)

1 – Binomcdf(n, p, x) Find (X > x) = 1 – P(X < x-1)

1 – Binomcdf(n, p, x-1)

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Page 7: Discrete Random Variables Chapter 4 Objectives 1

CHAPTER 5

Continuous Random VariablesChapter 5 Objectives

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Page 8: Discrete Random Variables Chapter 4 Objectives 1

CHAPTER 5 OBJECTIVES

The student will be able to

Recognize and understand continuous probability density functions in general.

Recognize the uniform probability distribution and apply it appropriately.

Recognize the exponential probability distribution and apply it appropriately.

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Page 9: Discrete Random Variables Chapter 4 Objectives 1

CONTINUOUS RANDOM

VARIABLES Types

UniformExponentialNormal

CharacteristicsOutcomes cannot be counted,

rather, they are measuredProbability is equal to an area

under the curve for the graph.Probability of exactly x is zero

since there is no area under the curve

PDF is a curve and can be drawn so we use f(x) to describe the curve, I.E. there is an equation for the curve

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Page 10: Discrete Random Variables Chapter 4 Objectives 1

UNIFORM DISTRIBUTION

X = a real number between a and b

X ~ U(a, b) µ = (a+b)/2 σ = sqrt((b-a)2/12) Probability density function

f(x) = 1/(b – a) To calculate probability find the

area of the rectangle under the curve P (X < x) = (x - a)*f(x) P (X > x) = (b – x)*f(x) P (c < X < d) = (d – c)*f(x) (we are not doing conditional

probability)

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Page 11: Discrete Random Variables Chapter 4 Objectives 1

UNIFORM DISTRIBUTION

Example - The amount of time a car must wait to get on the freeway at commute time is uniformly distributed in the interval from 1 to 6 minutes.

X = the amount of time (in minutes) a car waits to get on the freeway at commute time

1 < x < 6 X ~ U(1, 6)

µ = (6 + 1)/2 = 3.50

σ = sqrt((6 – 1)2/12) = 1.44

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Page 12: Discrete Random Variables Chapter 4 Objectives 1

UNIFORM DISTRIBUTION

What is the probability a car must wait less than 3 minutes? Draw the picture to solve the problem.P(X < 3) = ____________

P(2.5 < x < 5.6)

Find the 40th percentile.

The middle 60% is between what values?

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Page 13: Discrete Random Variables Chapter 4 Objectives 1

EXPONENTIAL DISTRIBUTION

X ~ Exp(m) X = a real number, 0 or larger. m = rate of decay or decline Mean and standard deviation

µ = σ = 1/m therefore m = 1/µ

PDF f (x) = me^(-mx)

Probability calculations P (X < x) = 1 – e^(-mx) P (X > x) = e^(-mx) P (c < X < d) = e^(-mc) – e^(-md)

Percentiles k = (LN(1-AreaToThe Left))/(-m)

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Page 14: Discrete Random Variables Chapter 4 Objectives 1

EXPONENTIAL DISTRIBUTION

An example - Count change. Calculate mean, standard

deviation and graphX = amount of change one

person carries0 < x < ?X ~ Exp( m )µ = σ = 1/ m

Find P(X < $2.50), P(X > $1.50), P($1.50 < X < $2.50), P(X < k) = 0.90

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Page 15: Discrete Random Variables Chapter 4 Objectives 1

CHAPTER 6

The Normal DistributionChapter 6 Objectives

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Page 16: Discrete Random Variables Chapter 4 Objectives 1

CHAPTER 6 OBJECTIVES

The student will be able to Recognize the normal

probability distribution and apply it appropriately.

Recognize the standard normal probability distribution and apply it appropriately.

Compare normal probabilities by converting to the standard normal distribution.

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Page 17: Discrete Random Variables Chapter 4 Objectives 1

THE NORMAL DISTRIBUTION

The Bell-shaped curve IQ scores, real estate prices,

heights, grades Notation

X ~ N(µ, σ )P(X < x), P(X > x), P(x1 < X <

x2)

Standard Normal Distributionz-score

Converts any normal distribution to a distribution with mean 0 and standard deviation 1

Allows us to compare two or more different normal distributionsz = (x - µ)/ σ

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Comparing

Page 18: Discrete Random Variables Chapter 4 Objectives 1

THE NORMAL DISTRIBUTION

Calculator Normalcdf(lowerbound,upperbound

,µ, σ) if P(X < x) then lowerbound is -

1E99 if P(X > x) then upperbound is

1E99percentiles

invNorm(percentile,µ,σ)

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example

Page 19: Discrete Random Variables Chapter 4 Objectives 1

CHAPTER 7

The Central Limit TheoremChapter 7 Objectives

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Page 20: Discrete Random Variables Chapter 4 Objectives 1

CHAPTER 7 OBJECTIVES

The student will be able to

Recognize the Central Limit Theorem problems.

Classify continuous word problems by their distributions.

Apply and interpret the Central Limit Theorem for Averages

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Page 21: Discrete Random Variables Chapter 4 Objectives 1

THE CENTRAL LIMIT THEOREM

Averages If we collect samples of size n

and n is “large enough”, calculate each sample’s mean and create a histogram of those means, the histogram will tend to have an approximate normal bell shape.

If we use X = mean of original random variable X, and X = standard deviation of original variable X then

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nNX

xx

,~

Page 22: Discrete Random Variables Chapter 4 Objectives 1

CENTRAL LIMIT THEOREM

Demonstration of concept Calculator

still use normalcdf and invnorm but need to use the correct standard deviation.

Normalcdf(lower, upper,X,X/sqrt(n))

Using the concept

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Page 23: Discrete Random Variables Chapter 4 Objectives 1

REVIEW FOR EXAM 2

What’s fair gameChapter 4Chapter 5Chapter 6Chapter 7

21 multiple choice questionsThe last 3 quarters’ exams

What to bring with youScantron (#2052), pencil,

eraser, calculator, 1 sheet of notes (8.5x11 inches, both sides)

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