Chapter 4
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Chapter 4. Discrete Random Variables Chapter 4 Objectives. 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.

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Chapter 4

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Chapter 4

Chapter 4

Discrete Random Variables

Chapter 4 Objectives


Chapter 4 objectives

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).


Discrete random variables

Discrete Random Variables

  • Types

    • General

    • Binomial

    • Poisson (not doing)

    • Geometric (not doing)

    • Hypergeometric (not doing)

  • Calculator becomes major tool


General discrete variables

General Discrete Variables

  • Probability Distribution Function (PDF)

    • Characteristics

      • each probability is between 0 and 1, inclusive

      • the sum of the probabilities is 1

    • An edit of the Relative Frequency Table where the RelFreq column is relabeled P(X) and we drop the Freq and Cum Freq columns

    • Calculated from the PDF

      • Mean (expected value)

      • Standard Deviation

An example


Binomial

Binomial

  • Characteristics

    • each probability is between 0 and 1, inclusive

    • the sum of the probabilities is 1

  • fixed number of trials

  • only 2 possible outcomes

  • for each trial, probabilities, p and q, remain the same (p + q = 1)

  • Other facts

    • X ~ B(n, p)

    • X = number of successes

    • n = number of independent trials

    • x = 0,1,2,…,n

    • µ = np

    • σ = sqrt(npq)

  • Problem 8


    Using calculator for binomial

    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)


    Chapter 5

    Chapter 5

    Continuous Random Variables

    Chapter 5 Objectives


    Chapter 5 objectives

    Chapter 5 Objectives

    The student will be able to

    • Recognize and understand continuous probability distribution functions in general.

    • Recognize the uniform probability distribution and apply it appropriately.

    • Recognize the exponential probability distribution and apply it appropriately.


    Continuous random variables

    Continuous Random Variables

    • Types

      • Uniform

      • Exponential

      • Normal

    • Characteristics

      • Outcomes cannot be counted, rather, they are measured

      • Probability 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


    Exponential distribution

    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) = P (X < d) - P (X < c) =

      1 - e^(-md) - (1 – e^(-mc) = e^(-mc) – e^(-md)

  • Percentiles

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


  • Exponential distribution1

    Exponential Distribution

    • An example - Count change.

    • Calculate mean, standard deviation and graph

      • X = amount of change one person carries

      • 0 < 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


    Uniform distribution

    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)


    Uniform distribution1

    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 = 3.5

      σ = sqrt((6 – 1)2/12) = 1.4434


    Uniform distribution2

    Uniform Distribution

    • What is the probability a car must wait less than 3 minutes? Draw the picture to solve the problem.

      • P(X < 3) = ____________


    Uniform distribution conditional probability

    Uniform DistributionConditional Probability

    • What is probability that a car waits more than 4 minutes given it has already waited more than 3 minutes?

      • P(X >4|X >3) = _____________


    Chapter 6

    Chapter 6

    The Normal Distribution

    Chapter 6 Objectives


    Chapter 6 objectives

    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.


    The normal distribution

    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 Distribution

      • z-score

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

        • Allows us to compare two or more different normal distributions

          • z = (x - µ)/ σ


    The normal distribution1

    The Normal Distribution

    • Calculator

      • Normalcdf(lowerbound,upperbound,µ, σ)

        • if P(X < x) then lowerbound is -1E99

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

      • percentiles

        • invNorm(percentile,µ, σ)


    Chapter 7

    Chapter 7

    The Central Limit Theorem

    Chapter 7 Objectives


    Chapter 7 objectives

    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


    The central limit theorem

    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


    Central limit theorem

    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


    Review for exam 2

    Review for Exam 2

    • What’s fair game

      • Chapter 4

      • Chapter 5

      • Chapter 6

      • Chapter 7

    • 21 multiple choice questions

      • The last 3 quarters’ exams

    • What to bring with you

      • Scantron (#2052), pencil, eraser, calculator, 1 sheet of notes (8.5x11 inches, both sides)


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