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Unformatted text preview: ution is d – c and the height is
1/(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
10 Example 5.1
Example Uniform Waiting Time #1 The amount of time, x, that a randomly selected hotel
The
that
patron spends waiting for the elevator at dinnertime.
patron
Record suggests x is uniformly distributed between zero
Record
and four minutes
and So c = 0 and d = 4,
So
1 f ( x ) = 4 −0 0 1
=
4 for 0 ≤ x ≤ 4 otherwise b −a
b −a
P( a ≤ x ≤ b ) =
=
4 −0
4
11 Uniform Waiting Time #2 What is the probability that a randomly selected hotel patron
What
waits at least 2.5 minutes for the elevator?
waits The probability is the area under the uniform distribution
The
in the interval [2.5, 4] minutes
in The probability is the area of a rectangle with
The
height ¼ and base 4 – 2.5 = 1.5
height P(x ≥ 2.5) = P(2.5 ≤ x ≤ 4) = ¼ × 1.5 = 0.375
(2.5 What is the probability that a randomly selected hotel patron
waits less than one minutes for the elevator? P(x ≤ 1) = P(0 ≤ x ≤ 1) = ¼ × 1 = 0.25
12 Uniform Waiting Time #3 Expect to wait the mean time µX µX 0 +4
=
= 2 minutes
2 with a standard deviation σX of
of
4 −0 σX = 12 =1.1547 minutes 13 Section5.3 The Normal Distribution
The normal probability distribution is defined by
The normal
the equation
the
f( x ) 2 f( x ) = 1
σ 2π e 1 x −µ − 2 σ µ for all values x on the real number line, where
µ is the mean and σ is the standard deviation,
π= 3.14159,
e = 2.71828 is the base of natural logarithms
14 x The normal curve is
symmetrical and bellshaped
• The normal is symmetrical
about its meanμ
• The mean is in the middle
under the curve
• So μ is also the median
• The normal is tallest over its
mean μ
• The area under the entire
normal curve is 1
• The area under either half
of the curve is 0.5
15 Properties of the Normal Distribution There is an infinite number of possible normal curves The tails of the normal extend to infinity in both
directions The particular shape of any individual normal depends
on its specific mean μ and standard deviation σ The tails get closer to the horizontal axis but never
touch it mean = median = mode The left and right halves of the curve are mirror
images of each other
16 The Position and Shape of the Normal Curve (a) The mean µ positions the peak of the normal curve
over the real axis
17 The Position and Shape of the Normal Curve (b) The variance σ2 measures the width or spread of the
normal curve
18 Normal Probabilities
Suppose x is a normally distributed random variable with
mean µ and standard deviation...
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 Spring '12
 vincent

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