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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

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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