Download Chapter 4: CONTINUOUS RANDOM VARIABLES AND

Chapter 4: CONTINUOUS RANDOM
VARIABLES AND
PROBABILITY DISTRIBUTIONS
Part 4:
Gamma Distribution
Weibull Distribution
Lognormal Distribution
Sections 4-9 through 4-11
Another exponential distribution example first...
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• Example: Magnitude of Earthquakes
The magnitude of earthquakes in a region
can be modeled as having an exponential distribution where the mean of the distribution
is 2.4, as measured on the Richter scale.
Let X represent the magnitude of an earthquake. Then, X ∼ exponential(λ).
First, determine λ:
1
E(X) = = 2.4 ⇒
λ
1
λ=
≈ 0.4167
2.4
In this problem, X is not a ‘wait time’ and is
not related to the Poisson process. Here, λ is
not a rate parameter, but is simply a parameter that tells you the shape of the distribution
of earthquake magnitudes.
2
f(x)
0
2
4
6
8
10
x
Find the probability that an earthquake striking this region will...
a) exceed 3.0 on the Richter scale.
b) fall between 2.0 and 3.0 on the Richter
scale.
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a) exceed 3.0 on the Richter scale.
b) fall between 2.0 and 3.0 on the Richter
scale.
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Gamma Distribution
Section 4-9
Another continuous distribution on x > 0 is the
gamma distribution.
• Gamma Distribution
The random variable X with probability density function
λr xr−1e−λx
f (x) =
for x > 0
Γ (r)
is a gamma random variable with parameters λ > 0 and r > 0.
• Mean and Variance
For a gamma random variable with parameters λ and r,
r
µ = E(X) =
λ
5
and
r
2
σ = V (X) = 2
λ
• The Gamma function: Γ (r)
The value in the denominator of f (x) is a
constant dependent on r. This value is:
Z ∞
Γ (r) =
xr−1e−xdx for r > 0
0
and this is a finite integral.
You can think of Γ (r) as a necessary constant in f (x) to make sure the area under
f (x) is 1.0
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The gamma family (expressed as choices of λ
and r) is very flexible:
0.6
0.8
Gamma distributions with fixed scale parameter (lambda=1)
0.4
0.0
0.2
f(x)
r=0.2
r=1
r=5
0
5
10
15
X
λ is called the scale parameter as it most influences the spread.
r is called the shape parameter as it most influences the peaked-ness of the distribution.
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• Example: Gene expression data
As technology progresses, so does the kind of
data we can collect. We can now gather information on the amount of protein (or mRNA)
outputted by a certain gene in an organism.
Below is a cDNA microarray slide showing
the amount of mRNA (as intensity of fluorescence) for each of thousands of genes for
a single organism.
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If we consider the expression values from many
individuals for a single gene:
0.3
0.0
0.1
0.2
Density
0.4
0.5
Gene 4 expression values
0
1
2
3
expression
We can model these expression values all coming from a specific gene with a gamma distribution...
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4
0.3
0.0
0.1
0.2
Density
0.4
0.5
Gene 4 expression values
0
1
2
3
expression
The solid curve is the ‘best fitting’ gamma
distribution to the observed data.
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4
Modeling the observed data with a common
distribution allows us to compute theoretical
probabilities, and compare different groups
(such as healthy patients vs. cancer patients).
——————————————————–
• Gamma distribution modeling
examples:
– Gene expression data
– Climatology models for monthly
precipitation
– The sum of k independent exponential random variables
The integration for gamma probabilities would
come from tables (like we saw for the normal distribution)... your book does not include these.
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Instead...
book homework problems are about recognizing
the gamma probability density function, setting
up f (x), and recognizing the mean µ and variance σ 2 (which can be computed from λ and r),
and seeing the connection of the gamma to the
exponential and the Poisson process.
• Example: The time between failures of a
laser machine is exponentially distributed with
a mean of 25,000 hours.
a) What is the expected time until the second
failure?
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13
b) What is the probability that the time until the third failure exceeds 50,000 hours?
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15
• Thus, we have another gamma
distribution modeling example:
– Time until rth failure in a Poisson Process with rate parameter λ is distributed
gamma(r, λ).
• Some comments on the gamma(r, λ)
distribution:
– When r = 1, f (x) is an exponential distribution with parameter λ. The exponential
distribution is a special case of the gamma
distribution.
– If r is a positive integer, the distribution
is called an Erlang distribution.
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– Some relationships:
Γ (1) = 0! = 1
Γ (r + 1) = rΓ (r)
{Γ (4) = 3·2·1 = 6}
Γ (r+1) = r!
√
1
Γ (2) = π
– For the gamma(r, λ) distribution, when
λ = 1/2 and r = p/2 where p is a positive integer, then we have a chi-squared
distribution with parameter p, another special case of the gamma.
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Weibull Distribution
Section 4-10
Another continuous distribution for x > 0.
It can be used to model a situation where the
number of failures increases with time, decreases
with time, or remains constant with time. (So,
it’s used for more complicated situations than a
Poisson process).
• Weibull Distribution
The random variable X with probability density function
β−1
β x
x β
f (x) =
exp −
for x > 0
δ δ
δ
is a Weibull random variable with scale parameter δ > 0 and shape parameter β > 0.
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• Mean and Variance
For a Weibull random variable with parameters β and δ,
1
µ = E(X) = δΓ 1 +
β
and
2
2
1
2
2
2
σ = V (X) = δ Γ 1 +
−δ Γ 1 +
β
β
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Also a flexible family:
20
• Cumulative Distribution Function
If X has a Weibull distribution with parameters δ and β, then the cumulative distribution function of X is
β
−( xδ )
F (x) = 1 − e
• A comment on the W eibull(δ, β) distribution:
– When β = 1, f (x) is an exponential distribution with parameter 1/δ. The exponential is a special case of the Weibull.
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• Example: Manufacture of semiconductor
In an industrial engineering article, the authors suggest using a Weibull distribution to
model the duration of a bake step in the manufacture of a semiconductor.
Let T represent the duration in hours of the
bake step for a randomly chosen lot.
Suppose T ∼ W eibull(δ = 10, β = 0.3).
a) What is the probability that the bake step
takes longer than four hours?
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b) What is the probability that the bake step
takes between two and seven hours?
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Lognormal Distribution
Section 4-11
The last continuous distribution we will consider
is also for x > 0.
Let W be a normally distributed random variable. Suppose we create a new random variable X with the transformation X = exp(W ).
Then, X is a lognormal random variable. The
name follows from the fact that
ln(X) = W
so we have ln(X) being normally distributed.
W can take on values from −∞ to ∞.
But the domain (or range) of X is the positive
real numbers.
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• Lognormal Distribution
Let W have a normal distribution with mean
θ and variance ω 2, then X = exp(W ) is a
lognormal random variable with probability
density function
"
#
1
(ln x − θ)2
√
f (x) =
exp −
for x > 0.
2
2ω
xω 2π
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• Mean and Variance
For a lognormal distribution with parameters
θ and ω 2,
2/2
θ+ω
µ = E(X) = e
and
σ 2 = V (X) = e
2θ+ω 2
e
ω2
−1
Note that for the lognormal r.v. X, the mean
and variance are µ and σ 2 and these are
functions of θ and ω 2, which are the mean
and variance of W , a normal random variable such that X = exp(W ).
So, θ and ω 2 show up in W ∼ N (θ, ω 2).
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• Example: Component lifetimes
Lifetimes of a randomly chosen component
are lognormally distributed with parameters
θ = 1 and ω = 0.5 days.
a) Find the mean lifetime of these components.
b) Find the standard deviation of the lifetimes.
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c) Find the probability that a component
lasts longer than 4 days.
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