Transcript Normal Distribution

Continuous Probability Distributions
F(x)
f(x)
x
Recall:
P(a < X < b) =
F (a) =
μ = E[X] =
s2 = E[X2] – μ2
x
= F(b) – F(a)
STAT509
f(x)
Continuous Uniform Distribution
1/(B-A)
x
f(x;A,B) = 1/(B-A), A < x < B
where [A,B] is an interval on the real
number line
μ = (A + B)/2 and s2 = (B-A)^2/12
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Continuous Uniform Distribution
• EX: Let X be the uniform continuous random
variable that denotes the current measured in a
copper wire in milliamperes. 0 < x < 50 mA. The
probability density function of X is uniform.
• f(x) =
, 0 < x < 50
• What is the probability that a measurement of
current is between 20 and 30 mA?
• What is the expected current passing through the
wire?
• What is the variance of the current passing
through the wire?
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Normal Distribution
f(x)
x
n(x;μ,σ) =
E[X] = μ
and
Var[X] = s2
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Normal Distribution
• Many phenomena in nature, industry and
research follow this bell-shaped
distribution.
– Physical measurements
– Rainfall studies
– Measurement error
• There are an infinite number of normal
distributions, each with a specified μ and s.
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Normal Distribution
• Characteristics
– Bell-shaped curve
– - < x < +
– μ determines distribution location and is the
highest point on curve
– Curve is symmetric about μ
– s determines distribution spread
– Curve has its points of inflection at μ + s
– μ + 1s covers 68% of the distribution
– μ + 2s covers 95% of the distribution
– μ + 3s covers 99.7% of the distribution
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Normal Distribution
s
s
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-3
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-1
μ
0
s
1
s
2
3
4
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Normal Distribution
n(x; μ = 0, s = 1)
n(x; μ = 5, s = 1)
f(x)
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-1
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x
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Normal Distribution
n(x; μ = 0, s = 0.5)
f(x)
n(x; μ = 0, s = 1)
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-3
-2
-1
0
1
2
3
4
x
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Normal Distribution
n(x; μ = 5, s = .5)
n(x; μ = 0, s = 1)
f(x)
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Normal Distribution
-4
-3
-2
-1
0
1
2
μ + 1s covers 68%
3
4
-4
-3
-2
-1
0
1
2
μ + 2s covers 95%
3
4
-4
-3
-2
-1
0
1
2
3
μ + 3s covers 99.7%
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4
Standard Normal Distribution
The distribution of a normal random variable
with mean 0 and variance 1 is called a
standard normal distribution.
-4
-3
-2
-1
0
1
2
3
4
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Standard Normal Distribution
• The letter Z is traditionally used to represent
a standard normal random variable.
• z is used to represent a particular value of Z.
• The standard normal distribution has been
tabularized.
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Standard Normal Distribution
Given a standard normal distribution, find the
area under the curve
(a) to the left of z = -1.85
(b) to the left of z = 2.01
(c) to the right of z = –0.99
(d) to right of z = 1.50
(e) between z = -1.66 and z = 0.58
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Standard Normal Distribution
Given a standard normal distribution, find the
value of k such that
(a) P(Z < k) = .1271
(b) P(Z < k) = .9495
(c) P(Z > k) = .8186
(d) P(Z > k) = .0073
(e) P( 0.90 < Z < k) = .1806
(f) P( k < Z < 1.02) = .1464
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Normal Distribution
• Any normal random variable, X, can be
converted to a standard normal random
variable:
z = (x – μx)/sx
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Normal Distribution
• Given a random Variable X having a normal
distribution with μx = 10 and sx = 2, find
the probability that X < 8.
z
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x
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-1
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Normal Distribution
• EX: The engineer responsible for a line that
produces ball bearings knows that the
diameter of the ball bearings follows a
normal distribution with a mean of 10 mm
and a standard deviation of 0.5 mm. If an
assembly using the ball bearings, requires
ball bearings 10 + 1 mm, what percentage
of the ball bearings can the engineer expect
to be able to use?
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Normal Distribution
•
EX: Same line of ball bearings.
(a) What is the probability that a randomly
chosen ball bearing will have a diameter less
than 9.75 mm?
(b) What percent of ball bearings can be expected
to have a diameter greater than 9.75 mm?
(c) What is the expected diameter for the ball
bearings?
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Normal Distribution
• EX: If a certain light bulb has a life that is
normally distributed with a mean of 1000
hours and a standard deviation of 50 hours,
what lifetime should be placed in a
guarantee so that we can expect only 5% of
the light bulbs to be subject to claim?
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Normal Distribution
• EX: A filling machine produces “16oz”
bottles of Pepsi whose fill are normally
distributed with a standard deviation of 0.25
oz. At what nominal (mean) fill should the
machine be set so that no more than 5% of
the bottles produced have a fill less than
15.50 oz?
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Relationship between the Normal and
Binomial Distributions
• The normal distribution is often a good
approximation to a discrete distribution when the
discrete distribution takes on a symmetric bell
shape.
• Some distributions converge to the normal as their
parameters approach certain limits.
• Theorem 6.2: If X is a binomial random variable
with mean μ = np and variance s2 = npq, then the
limiting form of the distribution of Z = (X –
np)/(npq).5 as n  , is the standard normal
distribution, n(z;0,1).
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Relationship between the Normal and
Binomial Distributions
• Consider b(x;15,0.4). Bars are calculated
from binomial. Curve is normal
approximation to binomial.
b(x;15,0.4)
0.25
f(x)
0.20
0.15
0.10
0.05
0.00
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Relationship between the Normal and
Binomial Distributions
• Let X be a binomial random variable with
n=15 and p=0.4.
• P(X=5) = ?
• Using normal approximation to binomial:
n(3.5<x<4.5;μ=6, s=1.897) = ?
Note, μ = np = (15)(0.4) = 6
s = (npq).5 = ((15)(0.4)(0.6)).5 = 1.897
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Relationship between the Normal and
Binomial Distributions
• EX: Suppose 45% of all drivers in a certain
state regularly wear seat belts. A random
sample of 100 drivers is selected. What is
the probability that at least 65 of the drivers
in the sample regularly wear a seatbelt?
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