Chapter 4

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# Chapter 4 - PowerPoint PPT Presentation

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

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.
• Calculate and interpret expected value (average).
Discrete Random Variables
• Types
• General
• Binomial
• Poisson (not doing)
• Geometric (not doing)
• Hypergeometric (not doing)
• Calculator becomes major tool
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
• 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
• 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

Continuous Random Variables

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

The Normal Distribution

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 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 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

The Central Limit Theorem

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
• 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
• 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
• 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)