Transcript Chapter 4: Random Variables and Probability Distributions

Chapter 5: Continuous Random
Variables
Where We’ve Been


Using probability rules to find the
probability of discrete events
Examined probability models for
discrete random variables
McClave: Statistics, 11th ed. Chapter 5: Continuous
Random Variables
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Where We’re Going



Develop the notion of a probability
distribution for a continuous random
variable
Examine several important continuous
random variables and their probability
models
Introduce the normal probability
distribution
McClave: Statistics, 11th ed. Chapter 5: Continuous
Random Variables
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5.1: Continuous Probability
Distributions
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A continuous random variable can
assume any numerical value within
some interval or intervals.
The graph of the probability distribution
is a smooth curve called a



probability density function,
frequency function or
probability distribution.
McClave: Statistics, 11th ed. Chapter 5:
Continuous Random Variables
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5.1: Continuous Probability
Distributions

There are an
infinite number of
possible outcomes


p(x) = 0
Instead, find
p(a<x<b)
 Table
 Software
 Integral calculus)
McClave: Statistics, 11th ed. Chapter 5:
Continuous Random Variables
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5.2: The Uniform Distribution
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
X can take on any
value between c and d
with equal probability
= 1/(d - c)
For two values a and b
ba
P(a  x  b) 
d c
cabd
McClave: Statistics, 11th ed. Chapter 5:
Continuous Random Variables
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5.2: The Uniform Distribution
Mean:
cd

2
Standard Deviation:
d c

12
McClave: Statistics, 11th ed. Chapter 5:
Continuous Random Variables
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5.2: The Uniform Distribution
Suppose a random variable x is
distributed uniformly with
c = 5 and d = 25.
What is P(10  x  18)?
McClave: Statistics, 11th ed. Chapter 5:
Continuous Random Variables
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5.2: The Uniform Distribution
Suppose a random variable x is
distributed uniformly with
c = 5 and d = 25.
What is P(10  x  18)?
18  10
P(10  x  18) 
 .40
25  5
McClave: Statistics, 11th ed. Chapter 5:
Continuous Random Variables
9
5.3: The Normal Distribution

Closely approximates many situations


Perfectly symmetrical around its mean
The probability density function f(x):
f ( x) 
1
e
 2
[( x   ) /  ]2

2
µ = the mean of x
 = the standard deviation of x
 = 3.1416…
e = 2.71828 …
McClave: Statistics, 11th ed. Chapter 5:
Continuous Random Variables
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5.3: The Normal Distribution


Each combination of µ and  produces a
unique normal curve
The standard normal curve is used in
practice, based on the standard normal
random variable z (µ = 0,  = 1), with the
probability distribution
1
f ( z) 
e
2
z2

2
The probabilities for z are given in Table IV
McClave: Statistics, 11th ed. Chapter 5:
Continuous Random Variables
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5.3: The Normal Distribution
P(0  z  1.00)  .3413
P(1.00  z  0)  .3413
P(1  z  1)
 .3413 .3413
 .6826
P(1  z  1.25) 
P(0  z  1.25)  P(0  z  1.00)
 .3944 .3413  .0531
McClave: Statistics, 11th ed. Chapter 5:
Continuous Random Variables
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5.3: The Normal Distribution
For a normally
distributed random
variable x, if we know
µ and ,
zi 
xi  

So any normally
distributed variable
can be analyzed
with this single
distribution
McClave: Statistics, 11th ed. Chapter 5:
Continuous Random Variables
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5.3: The Normal Distribution
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
Say a toy car goes an average of 3,000 yards between
recharges, with a standard deviation of 50 yards (i.e., µ
= 3,000 and  = 50)
What is the probability that the car will go more than
3,100 yards without recharging?
McClave: Statistics, 11th ed. Chapter 5:
Continuous Random Variables
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5.3: The Normal Distribution


Say a toy car goes an average of 3,000 yards between
recharges, with a standard deviation of 50 yards (i.e., µ
= 3,000 and  = 50)
What is the probability that the car will go more than
3,100 yards without recharging?
3100 3000

P( x  3100)  P z 

50


P( z  2.00)  1  P( z  2.00) 
1  .5  P(0  z  2.00) 
1  .5  .4772 .0228
McClave: Statistics, 11th ed. Chapter 5:
Continuous Random Variables
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5.3: The Normal Distribution

To find the probability for a normal random
variable …
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



Sketch the normal distribution
Indicate x’s mean
Convert the x variables into z values
Put both sets of values on the sketch, z below x
Use Table IV to find the desired probabilities
McClave: Statistics, 11th ed. Chapter 5:
Continuous Random Variables
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5.4: Descriptive Methods for
Assessing Normality
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If the data are normal
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A histogram or stem-and-leaf display will look like
the normal curve
The mean ± s, 2s and 3s will approximate the
empirical rule percentages
The ratio of the interquartile range to the standard
deviation will be about 1.3
A normal probability plot , a scatterplot with the
ranked data on one axis and the expected z-scores
from a standard normal distribution on the other
axis, will produce close to a straight line
McClave: Statistics, 11th ed. Chapter 5:
Continuous Random Variables
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5.4: Descriptive Methods for
Assessing Normality

IQR 22

 1.29
s
17
Errors per MLB team in 2003
Mean: 106
Standard Deviation: 17

IQR: 22



Frequency
Histogram
10
9
8
7
6
5
4
3
2
1
0
x  2s  106 34
72  140 28 out of 30: 93%
Frequency
77
89.8 102.6 115.4 128.2 More
Errors per team, 2003

x  s  106 17
89  123 22 out of 30: 73%
x  3s  106 51
55  157 30 out of 30: 100%
McClave: Statistics, 11th ed. Chapter 5:
Continuous Random Variables
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5.4: Descriptive Methods for
Assessing Normality

3
Normal Quantile
2
1
0
A normal probability
plot is a scatterplot with
the ranked data on one
axis and the expected zscores from a standard
normal distribution on
the other axis
-1
-2
-3
60
80
100
120
140
160
Errors
McClave: Statistics, 11th ed. Chapter 5:
Continuous Random Variables
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5.5: Approximating a Binomial
Distribution with the Normal
Distribution


Discrete calculations may become very
cumbersome
The normal distribution may be used to
approximate discrete distributions

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The larger n is, and the closer p is to .5, the
better the approximation
Since we need a range, not a value, the
correction for continuity must be used

A number r becomes r+.5
McClave: Statistics, 11th ed. Chapter 5:
Continuous Random Variables
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5.5: Approximating a Binomial
Distribution with the Normal
Distribution
Calculate the mean plus/minus 3 standard deviations
  3  np npq
If this interval is in the range 0 to n,
the approximation will be reasonably close
Express the binomial probability as a range of values
P( x  a)
P ( x  b)  P ( x  a )
Find the z-values for each binomial value
z
(a  .5)  

Use the standard normal distribution to find
the probability for the range of values you calculated
McClave: Statistics, 11th ed. Chapter 5:
Continuous Random Variables
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5.5: Approximating a Binomial
Distribution with the Normal
Distribution
Flip a coin 100 times and compare the binomial
and normal results
Binomial:
Normal:
100 50 50
.5 .5  .0796
P( x  50)  
 50 
  100 .5  50
  100 .5  .5  5
50.5  50 
 49.5  50
P(49.5  x  50.5)  P
z

5
5


P(0.10  z  0.10)  .0796
McClave: Statistics, 11th ed. Chapter 5:
Continuous Random Variables
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5.5: Approximating a Binomial
Distribution with the Normal
Distribution
Flip a weighted coin [P(H)=.4] 10 times and
compare the results
Binomial:
Normal:
10 5 5
P( x  5)   .4 .6  .1204
5
  10  .4  4
  10  .4  .6  1.55
5.5  4 
 4.5  4
P(4.5  x  5.5)  P
z

1.55 
 1.55
P(0.32  z  0.32)  .1255
McClave: Statistics, 11th ed. Chapter 5:
Continuous Random Variables
23
5.5: Approximating a Binomial
Distribution with the Normal
Distribution
Flip a weighted coin [P(H)=.4] 10 times and
compare the results
Binomial:
Normal:
10 5 5
P( x  5)   .4 .6  .1204
5
The more p differs from .5,
  10  .4  4
and the smaller n is,
  10  .4  .6  1.55
the less precise the
approximation will be
5.5  4 
 4.5  4
P(4.5  x  5.5)  P
z

1.55 
 1.55
P(0.32  z  0.32)  .1255
McClave: Statistics, 11th ed. Chapter 5:
Continuous Random Variables
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5.6: The Exponential
Distribution

Probability Distribution for an Exponential
Random Variable x

Probability Density Function
f ( x) 
1

e  x /
( x  0)

Mean: µ = 

Standard Deviation:  = 
McClave: Statistics, 11th ed. Chapter 5:
Continuous Random Variables
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5.6: The Exponential
Distribution
Suppose the waiting time to see the nurse at the student
health center is distributed exponentially with a mean of
45 minutes. What is the probability that a student will
wait more than an hour to get his or her generic pill?
60
McClave: Statistics, 11th ed. Chapter 5:
Continuous Random Variables
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5.6: The Exponential
Distribution
Suppose the waiting time to see the nurse at the student
health center is distributed exponentially with a mean of
45 minutes. What is the probability that a student will
wait more than an hour to get his or her generic pill?
P( x  a)  e

P( x  60)  e
a


60
45
 e 1.33  .2645
60
McClave: Statistics, 11th ed. Chapter 5:
Continuous Random Variables
27