discrete random variables chapter 4 objectives 1
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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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DISCRETE RANDOM VARIABLES
TypesGeneralBinomialPoisson (not doing)Geometric (not doing)Hypergeometric (not doing)
Calculator becomes major tool
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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
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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
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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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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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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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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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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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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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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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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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CHAPTER 6
The Normal DistributionChapter 6 Objectives
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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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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
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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
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CHAPTER 7
The Central Limit TheoremChapter 7 Objectives
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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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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
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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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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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