Weibull Distribution
Definition
A random variable X is said to have a Weibull distribution with
parameters α and β (α > 0, β > 0) if the pdf of X is
(
α α−1 −(x/β)α
e
x ≥0
αx
f (x; α, β) = β
0
x <0
Remark:
1. The family of Weibull distributions was introduced by the Swedish
physicist Waloddi Weibull in 1939.
2. We use X ∼ WEB(α, β) to denote that the rv X has a Weibull
distribution with parameters α and β.
Liang Zhang (UofU)
Applied Statistics I
July 1, 2008
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Weibull Distribution
Remark:
3. When α = 1, the pdf becomes
(
f (x; β) =
1 −x/β
βe
x ≥0
0
x <0
which is the pdf for an exponential distribution with parameter λ = β1 .
Thus we see that the exponential distribution is a special case of both the
gamma and Weibull distributions.
4. There are gamma distributions that are not Weibull distributios and
vice versa, so one family is not a subset of the other.
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Applied Statistics I
July 1, 2008
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Weibull Distribution
Liang Zhang (UofU)
Applied Statistics I
July 1, 2008
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Weibull Distribution
Liang Zhang (UofU)
Applied Statistics I
July 1, 2008
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Weibull Distribution
Proposition
Let X be a random variable such that X ∼ WEI(α, β). Then
( 2 )
2
1
1
and V (X ) = β 2 Γ 1 +
− Γ 1+
E (X ) = βΓ 1 +
α
α
α
The cdf of X is
(
α
1 − e −(x/β)
F (x; α, β) =
0
Liang Zhang (UofU)
Applied Statistics I
x ≥0
x <0
July 1, 2008
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Weibull Distribution
Example:
The shear strength (in pounds) of a spot weld is a Weibull distributed
random variable, X ∼ WEB(400, 2/3).
a. Find P(X > 410).
b. Find P(X > 410 | X > 390).
c. Find E (X ) and V (X ).
d. Find the 95th percentile.
Liang Zhang (UofU)
Applied Statistics I
July 1, 2008
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Weibull Distribution
In practical situations, γ = min(X ) > 0 and X − γ has a Weibull
distribution.
Example (Problem 74):
Let X = the time (in 10−1 weeks) from shipment of a
defective product until the customer returns the product.
Suppose that the minimum return time is γ = 3.5 and that the excess
X − 3.5 over the minimum has a Weibull distribution with parameters
α = 2 and β = 1.5.
a. What is the cdf of X ?
b. What are the expected return time and variance of return time?
c. Compute P(X > 5).
d. Compute P(5 ≤ X ≤ 8).
Liang Zhang (UofU)
Applied Statistics I
July 1, 2008
7 / 15
Lognormal Distribution
Definition
A nonnegative rv X is said to have a lognormal distribution if the rv
Y = ln(X ) has a normal distribution. The resulting pdf of a lognormal rv
when ln(X ) is normally distributed with parameters µ and σ is
(
2
2
√ 1
e −[ln(x)−µ] /(2σ ) x ≤ 0
2πσx
f (x; µ, σ) =
0
x <0
Remark:
1. We use X ∼ LOGN(µ, σ 2 ) to denote that rv X have a lognormal
distribution with parameters µ and σ.
2. Notice here that the parameter µ is not the mean and σ 2 is not the
variance, i.e.
µ 6= E (X ) and σ 2 6= V (X )
Liang Zhang (UofU)
Applied Statistics I
July 1, 2008
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Lognormal Distribution
Liang Zhang (UofU)
Applied Statistics I
July 1, 2008
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Lognormal Distribution
Proposition
If X ∼ LOGN(µ, σ 2 ), then
E (X ) = e µ+σ
2 /2
2
2
and V (X ) = e 2µ+σ · (e σ − 1)
The cdf of X is
F (x; µ, σ) = P(X ≤ x) = P[ln(X ) ≤ ln(x)]
ln(x) − µ
ln(x) − µ
=P Z ≤
=Φ
σ
σ
x ≤0
where Φ(z) is the cdf of the standard normal rv Z .
Liang Zhang (UofU)
Applied Statistics I
July 1, 2008
10 / 15
Lognormal Distribution
Example (Problem 115)
Let Ii be the input current to a transistor and I0 be the output current.
Then the current gain is proportional to ln(I0 /Ii ). Suppose the constant of
proportionality is 1 (which amounts to choosing a particular unit of
measurement), so that current gain = X = ln(I0 /Ii ). Assume X is
normally distributed with µ = 1 and σ = 0.05.
a. What is the probability that the output current is more than twice the
input current?
b. What are the expected value and variance of the ratio of output to
input current?
c. What value r is such that only 5% chance we will have the ratio of
output to input current exceed r ?
Liang Zhang (UofU)
Applied Statistics I
July 1, 2008
11 / 15
Beta Distribution
Definition
A random variable X is said to have a beta distribution with parameters
α, β(both positive), A, and B if the pdf of X is

α−1 β−1
 1
Γ(α+β)
x−A
B−x
·
·
A≤x ≤B
·
B−A
B−A
f (x; α, β, A, B) = B−A Γ(α)·Γ(β)
0
otherwise
The case A = 0, B = 1 gives the standard beta distribution.
Remark: We use X ∼ BETA(α, β, A, B) to denote that rv X has a beta
distribution with parameters α, β, A, and B.
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Applied Statistics I
July 1, 2008
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Beta Distribution
Proposition
If X ∼ BETA(α, β, A, B), then
E (X ) = A + (B − A) ·
Liang Zhang (UofU)
α
(B − A)2 αβ
and V (X ) =
α+β
(α + β)2 (α + β + 1)
Applied Statistics I
July 1, 2008
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Beta Distribution
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Applied Statistics I
July 1, 2008
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Beta Distribution
Example (Problem 127)
An individual’s credit score is a number calculated based on that person’s
credit history which helps a lender determine how much he/she should be
loaned or what credit limit should be established for a credit card. An
article in the Los Angeles Times gave data which suggested that a beta
distribution with parameters A = 150, B = 850, α = 8, β = 2 would
provide a reasonable approximation to the distribution of American credit
scores. [Note: credit scores are integer-valued].
a. Let X represent a randomly selected American credit score. What are
the mean value and standard deviation of this random variable? What
is the probability that X is within 1 standard deviation of its mean
value?
b. What is the approximate probability that a randomly selected score
will exceed 750 (which lenders consider a very good score)?
Liang Zhang (UofU)
Applied Statistics I
July 1, 2008
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