Download Introduction to the Normal Distribution

Continuous Probability
Distributions
Chapter 4
The Normal Distribution
1
Introduction
A company that makes hot chocolate mix
has a machine packages the mixes. The
amount in each package varies slightly
from the nominal amount. For instance,
the 16 oz containers actually contain
between 15.95 and 17 oz of hot chocolate.
If the discrepancies are distributed
uniformly over the interval, what is the
probability that a package is underfilled?
The Normal Distribution
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The Uniform Distribution
A very simple
continuous probability
distribution is the
uniform distribution. If
a random variable X
varies uniformly
between a and b, then
X~U(a,b). The pdf of X
is flat, as shown at the
right.
The Normal Distribution
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The Uniform Distribution
What are f(x), F(x), E(X), and Var(X)?
What is the probability that the hot
chocolate filler machine underfills the
16oz container?
The Normal Distribution
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Introduction to the Normal
Distribution
Chapter 5
The Normal Distribution
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Introduction
MENSA is an organization that “provides
intelligent individuals an opportunity to meet
other smart people at the local, regional, and
national levels.” The MENSA website does not
specify what IQ score a person must obtain to be
admitted, instead it states: “The society
welcomes people from every walk of life whose
IQ is in the top 2% of the population”.
What IQ score must I have to be admitted to
MENSA?
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Introduction
Average IQ is about 100 points
Standard deviation is about 15 points
IQ is approximately normally distributed
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The Normal Distribution
A normally distributed random variable (also called
Gaussian) X~N(μ,σ2) has pdf
1
f ( x) =
e
2π σ
−( x−μ )2
2σ 2
for -∞≤x≤∞.
E(X)=μ
Var(X)=σ2
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The Normal Distribution
The normal distribution has very broad applicability in
part because
• Phenomena in the natural world that result from the
interaction of many environmental and genetic factors
tend to follow the normal distribution (e.g., height,
weight, measurable intelligence).
• The sums and averages of random samples have
distributions that look roughly normal – as the sample
size gets larger, the normal approximation gets better.
This result is known as the Central Limit Theorem.
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The Normal Distribution
The Standard Normal
Distribution: N(0,1)
1
f ( x) = φ ( x) =
e
2π
− x2
2
x
F ( x) = Φ ( x) = ∫ φ ( y )dy
−∞
In general, if X~N(μ,σ2), then Z=(X- μ)/σ ~ N(0,1).
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The Normal Distribution
Z=(X- μ)/σ ~ N(0,1) is a “standardized” version of the
random variable X.
The probability values of a general normal distribution
can be related to the cumulative distribution function of
the standard normal distribution Φ(X) as follows:
⎛b−μ ⎞
⎛a−μ⎞
P ( a ≤ X ≤ b) = Φ⎜
⎟ − Φ⎜
⎟
⎝ σ ⎠
⎝ σ ⎠
Table I in the book can be used to calculate probability
values of the standard normal distribution.
The Normal Distribution
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The Normal Distribution
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The Normal Distribution
If X~N(2.5 , 9) and Z~N(0, 1), find the
following probabilities:
–
–
–
–
P(Z ≤ 1)
P(X ≥ 7)
P(1.4 ≤ Z ≤ 2.5)
P(|X|≤ 3)
What is the probability that a randomly
selected individual has an IQ less than 70?
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The Normal Distribution
Table I can also be used to find percentiles of the standard
normal distribution.
Example: If X~N(2.5 , 9) and Z~N(0, 1), find the value of x
or z that satisfies each equation.
– P(X ≤ x) = .45
– P(Z ≥ z) = .39
– P(|Z| ≥ z ) = .72
What is the minimum score a person must obtain to be
admitted to MENSA? (Recall: MENSA admits the top 2% of the
population).
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The Normal Distribution
The length of human pregnancies from conception
to birth varies according to a distribution that is
approximately normal with mean 266 days and
standard deviation 16 days.
What is the probability that a pregnancy lasts less
than 240 days (about 8 months)?
What is c for which there is a 99% probability that
a pregnancy lasts longer than c days?
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The Normal Distribution
The probability that a
normal random variable
takes a value within 1
standard deviation of its
mean is about 68%, within 2
standard deviations is about
95%, and within 3 standard
deviations is about 99.7%.
P(1 ≤ Z ≤ 1) ≈ .68
P(2 ≤ Z ≤ 2) ≈ .95
P(3 ≤ Z ≤ 3) ≈ .997
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The Normal Distribution
For α<0.5, the (1- α)x100th percentile of the
distribution is denoted by Zα, such that Φ(Zα)=1-α.
These percentiles, Zα, are sometimes known as
“critical points”.
There is a probability of 1-α that a N(μ,σ2) random
variable takes on a value within the interval [μ-σ
Zα/2, μ+σ Zα/2].
We know from the previous example that if X=“the
length of a human pregnancy from conception to
birth”, then X~N(266, 256). Since Z0.005 = 2.576, we
can be 99% confident that the duration of the
pregnancy of a randomly selected woman will be in
the interval [μ-σ Z0.005, μ+σ Z0.005].
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Combining Normal R.V.s
We can use combinations of normal random
variables to address more complicated
questions such as:
The mean pregnancy length, 266 days, is
about .73 years. What is the probability that
a randomly selected woman who gives
birth 4 times will spend more than three
years of her life (1095 days) pregnant?
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Combining Normal R.V.s
If X is normally distributed, then a linear function of X also
has a normal distribution.
•If a and b are constants and X~N(μ,σ2) and Y=aX+b, then
Y~N(aμ+b, a2σ2).
•If X1~N(μ1,σ12) and X2~N(μ2,σ22) are independent random
variables, then Y=X1+X2~N(μ1+ μ2,σ12+ σ22). [This can be extended
to n independent normal random variables.]
•If Xi~N(μ,σ2), 1≤ i ≤n, are independent random variables,
⎛ σ2 ⎞
then
⎟⎟
X ~ N ⎜⎜ μ ,
n ⎠
⎝
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Combining Normal R.V.s
Example: Suppose the amount of soda pop in a ‘16
oz’ bottle is normally distributed with mean 16
oz and standard deviation 0.25 and that the
amounts in the bottles are independent.
What is the probability that there will be less than
95 oz of soda in 6 bottles?
What are the mean and the standard deviation of
the average amount of soda pop in 6 bottles?
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Combining Normal R.V.s
An animal shelter has a special facility for sick
animals. Suppose there are 3 cats and 2 dogs in
the shelter and all at risk for either “cat flu” or
“dog flu”. Let X=“the duration of cat flu
(hours)” and Y=“the duration of dog flu
(hours)”. If X~N(36,9) and Y~N(41, 6.25):
~ What is the distribution of Z=“the total number
of pet days spent in the infirmary”?
~ What is the probability that the average time a
cat spends in the infirmary is more than 40
hours?
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Approximating Distributions with
the Normal Distribution
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Approximating Distributions with
the Normal Distribution
Recall: if X~B(n,p), then E(X)=np and
Var(X)=np(1-p).
A binomial distribution, B(n,p), can be
approximated by a normal distribution
with the same mean and variance,
N(np,np(1-p)).
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Approximating Distributions with
the Normal Distribution
A continuity correction is used to improve the
approximation.
If X~B(np,np(1-p)), then
⎛ x + 0.5 − np ⎞
⎟
P ( X ≤ x ) ≈ Φ⎜
⎜ np(1 − p) ⎟
⎝
⎠
and
⎛ x − 0.5 − np ⎞
⎟
P ( X ≥ x ) ≈ 1 − Φ⎜
⎜ np(1 − p) ⎟
⎝
⎠
These approximations work well for np≥5 and n(1-p) ≥5.
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Approximating Distributions with
the Normal Distribution
Example: If X~B(14, .4), find P(X ≤ 3) exactly
and by using the normal approximation.
Example: In a test for a particular illness, a
false-positive result is obtained about 1 in
125 administrations of the test. If the test
is administered to 15,000 people, estimate
the probability of there being more than
135 false-positive results.
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Approximating Distributions with
the Normal Distribution
The Central Limit Theorem: If X1,…Xn, is a
sequence of independent identically distributed
random variables with a mean μ and variance σ2,
then the distribution of their average is
approximately normal
⎛ σ2 ⎞
X ~ N ⎜⎜ μ , ⎟⎟
n ⎠
⎝
The distribution of the sum X=X1+…+Xn is also
approximately normal
(
X ~ N nμ , nσ
The Normal Distribution
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)
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Approximating Distributions with
the Normal Distribution
Example: The number of cracks in a ceramic
tile has a Poisson distribution with
parameter λ=2.4. Use the normal
approximation to find the probability that
there are more than 1250 cracks in 500
ceramic tiles.
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Distributions Related to Normal
The Chi-Square
Distribution:
If X ~ N(0,1) then Y = X2
has a chi-square
distribution with 1
degree of freedom.
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Distributions Related to Normal
A t-distribution with ν
degrees of freedom is
defined as follows:
tν =
N (0,1)
χν /ν
2
where the N(0,1) and Xv2 are
independently distributed.
As v→∞, the t-distribution
tends toward a standard
normal distribution.
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Distributions Related to Normal
The F-distribution is
defined as the ratio of
two independent chisquare random
variables divided by
their respective
degrees of freedom:
Fν 1ν 2 ~
χν
χν
2
ν1
2
ν2
1
2
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