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Chapter 4

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Chapter 4

Discrete Random Variables

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).

- Types
- General
- Binomial
- Poisson (not doing)
- Geometric (not doing)
- Hypergeometric (not doing)

- Calculator becomes major tool

- 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

- Characteristics

An example

- 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)

- 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

- What the calculator can do
- Find P(X = x)
- Binompdf(n, p, x)

- Find P(X < x)
- Binomcdf(n, p, x)

- Find P(X = 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)

- Find P(X < x) = P(X < x-1)

Chapter 5

Continuous Random Variables

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.

- 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

- 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

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

- 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)

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

- 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

- 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)

- 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

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

- 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

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 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

- z-score

- 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)

- Normalcdf(lowerbound,upperbound,µ, σ)

Chapter 7

The Central Limit Theorem

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

- 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

- 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

- 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)