Chap006 New

Continuous Random Variables

Chapter 6

McGraw-Hill/Irwin Copyright 2011 by The McGraw-Hill Companies, Inc. All rights reserved.

Continuous RandomVariables

6.1 Continuous Probability Distributions6.2 The Uniform Distribution6.3 The Normal Probability Distribution6.4 Approximating the Binomial Distribution by

Using the Normal Distribution (Optional)6.5 The Exponential Distribution (Optional)6.6 The Normal Probability Plot (Optional)

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6.1 ContinuousProbability Distributions

A continuous random variable may assumeany numerical value in one or more intervals

Use a continuous probability distribution toassign probabilities to intervals of values

LO 1: Explain what acontinuous probabilitydistribution is and how itis used.

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Continuous ProbabilityDistributions Continued

The curve f(x) is the continuous probabilitydistribution of the random variable x if theprobability that x will be in a specified intervalof numbers is the area under the curve f(x)corresponding to the interval

Other names for a continuous probabilitydistribution: Probability curve Probability density function

LO1

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Properties of ContinuousProbability Distributions

Properties of f(x): f(x) is a continuousfunction such that

1. f(x) > 0 for all x2. The total area under the f(x) curve is equal to 1

Essential point: An area under a continuousprobability distribution is a probability

LO1

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Area and Probability

The blue area under the curve f(x) from x = ato x = b is the probability that x could takeany value in the range a to b Symbolized as P(a x b) Or as P(a < x < b), because each of the interval

endpoints has a probability of 0

LO1

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Distribution Shapes

Symmetrical and rectangular The uniform distribution (Section 6.2, Fig 6.2)

Symmetrical and bell-shaped The normal distribution (Section 6.3, Fig 6.1)

Skewed Skewed either left or right (Section 6.5)

LO1

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6.2 The UniformDistribution

If c and d are numbers on the real line (c < d), theprobability curve describing the uniform distribution is

The probability that x is any value between the values aand b (a < b) is

Note: The number ordering is c < a < b < d

dx ccd=x

otherwise0

for1f

cdabbxaP

LO 2: Use the uniformdistribution to computeprobabilities.

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The Uniform Distribution Continued The mean X and standard deviation X of a

uniform random variable x are

These are the parameters of the uniformdistribution with endpoints c and d (c < d)

12

2cd

dc

X

X

LO2

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The Uniform ProbabilityCurve

LO2

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Notes on the UniformDistribution

The uniform distribution is symmetrical Symmetrical about its center X X is the median

The uniform distribution is rectangular For endpoints c and d (c < d) the width of the distribution is

d c and the height is1/(d c)

The area under the entire uniform distribution is 1 Because width height = (d c) [1/(d c)] = 1 So P(c x d) = 1

LO2

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6.3 The NormalProbability Distribution

The normal probability distribution is defined by theequation

for all values x on the real number line is the mean and is the standard deviation = 3.14159 and e = 2.71828 is the base of natural

logarithms

21

=)f(2

21

e

x

x

LO 3: Describe theproperties of the normaldistribution and use acumulative normal table.

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The Normal ProbabilityDistribution Continued

The normal curve is symmetricalabout its mean The mean is in the middle under the

curve So is also the median

It is tallest over its mean The area under the entire normal

curve is 1 The area under either half of the curve

is 0.5

LO3

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Properties of the NormalDistribution

There are an infinite number of normal curves The shape of any individual normal curve depends on its

specific mean and standard deviation The highest point is over the mean Mean = median = mode

All measures of central tendency equal each other The only probability distribution for which this is true

The curve is symmetrical about its mean The left and right halves of the curve are mirror images

LO3

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Properties of the NormalDistribution Continued

The tails of the normal extend to infinity inboth directions The tails get closer to the horizontal axis but

never touch it The area under the normal curve to the right

of the mean equals the area under thenormal to the left of the mean The area under each half is 0.5

LO3

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The Position and Shape ofthe Normal Curve

(a) The mean positions the peak of the normal curve over the real axis(b) The variance 2 measures the width or spread of the normal curve

LO3

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Normal Probabilities Suppose x is a normally distributed

random variable with mean andstandard deviation

The probability that x could take anyvalue in the range between twogiven values a and b (a < b) is P(a x b)

P(a x b) is the area colored inblue under the normal curve andbetween the values x = a and x = b

LO3

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Three ImportantPercentages

LO3

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The Standard NormalDistribution

If x is normally distributed with mean andstandard deviation , then the randomvariable z

is normally distributed with mean 0 andstandard deviation 1

This normal is called the standard normaldistribution

xz

LO3

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The Standard NormalDistribution Continued

z measures the number of standard deviations that x is fromthe mean The algebraic sign on z indicates on which side of is x z is positive if x > (x is to the right of on the number line) z is negative if x < (x is to the left of on the number line)

LO3

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The Standard Normal Table

The standard normal table is a table that liststhe area under the standard normal curve tothe right of the mean (z=0) up to the z valueof interest See Table 6.1, Table A.3 in Appendix A, and the

table on the back of the front cover This table is so important that it is repeated 3 times in

the textbook! Always look at the accompanying figure for guidance

on how to use the table

LO3

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The Standard Normal TableContinued

The values of z (accurate to the nearesttenth) in the table range from 0.00 to 3.09 inincrements of 0.01 z accurate to tenths are listed in the far left

column The hundredths digit of z is listed across the top

of the table The areas under the normal curve to the right

of the mean up to any value of z are given inthe body of the table

LO3

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A Standard Normal Table

LO3

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The Standard NormalTable Example

Find P(0 z 1) Find the area listed in the table corresponding to a z value

of 1.00 Starting from the top of the far left column, go down to 1.0 Read across the row z = 1.0 until under the column headed

by .00 The area is in the cell that is the intersection of this row

with this column As listed in the table, the area is 0.3413, so

P(0 z 1) = 0.3413

LO 4: Use the normaldistribution to computeprobabilities.

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Calculating P(-2.53 z 2.53) First, find P(0 z 2.53) Go to the table of areas under the standard

normal curve Go down left-most column for z = 2.5 Go across the row 2.5 to the column headed by

.03 The area to the right of the mean up to a value of

z = 2.53 is the value contained in the cell that isthe intersection of the 2.5 row and the .03 column

The table value for the area is 0.4943

Continued

LO4

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Calculating P(-2.53 z 2.53)Continued

From last slide, P(0 z 2.53)=0.4943 By symmetry of the normal curve, this is also

the area to the left of the mean down to avalue of z = 2.53

Then P(-2.53 z 2.53) = 0.4943 + 0.4943 =0.9886

LO4

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Calculating P(z > -1)

An example of finding the area under thestandard normal curve to the right of anegative z value

Shown is finding the under the standardnormal for z -1

LO4

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Calculating P(z 1)

LO4

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Finding NormalProbabilities

1. Formulate the problem in terms of x values2. Calculate the corresponding z values, and restate

the problem in terms of these z values3. Find the required areas under the standard normal

curve by using the table

Note: It is always useful to draw a picture showingthe required areas before using the normal table

LO 5: Find populationvalues that correspondto specified normaldistribution probabilities.

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Finding Z Points on aStandard Normal Curve

LO5

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Finding a Tolerance Interval

Finding a tolerance interval [ k] thatcontains 99% of the measurements in anormal population

LO5

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6.4 Approximating the BinomialDistribution by Using theNormal Distribution (Optional)

The figure below shows several binomialdistributions

Can see that as n gets larger and as p gets closer to0.5, the graph of the binomial distribution tends tohave the symmetrical, bell-shaped, form of thenormal curve

LO 6: Use the normaldistribution toapproximate binomialprobabilities (optional).

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Normal Approximation tothe Binomial Continued

Generalize observation from last slide forlarge p

Suppose x is a binomial random variable,where n is the number of trials, each having aprobability of success p Then the probability of failure is 1 p

If n and p are such that np 5 andn(1p) 5, then x is approximately normalwith

pnpnp 1and

LO6

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6.5 The ExponentialDistribution (Optional)

Suppose that some event occurs as a Poissonprocess That is, the number of times an event occurs is a Poisson

random variable Let x be the random variable of the interval between

successive occurrences of the event The interval can be some unit of time or space

Then x is described by the exponential distribution With parameter , which is the mean number of events that

can occur per given interval

LO 7: Use theexponential distributionto compute probabilities(optional).

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The Exponential DistributionContinued

If is the mean number of events per giveninterval, then the equation of the exponentialdistribution is

The probability that x is any value betweengiven values a and b (a

The Exponential Distribution#3

The mean X and standard deviation X of anexponential random variable x are

1and1 XX

LO7

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6.6 The NormalProbability Plot

A graphic used to visually check to see ifsample data comes from a normal distribution

A straight line indicates a normal distribution The more curved the line, the less normal the

data is

LO 8: Use a normalprobability plot to helpdecide whether datacome from a normaldistribution (optional).

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Creating a NormalProbability Plot

1. Rank order the data from smallest to largest2. For each data point, compute the value i/(n+1)

i is the data points position on the list

3. For each data point, compute the standardizednormal quantile value (Oi)

Oi is the z value that gives an area i/(n+1) to its left

4. Plot data points against Oi5. Straight line indicates normal distribution

LO8

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