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ST 380
Probability and Statistics for the Physical Sciences
Other Continuous Distributions
Weibull Distribution
Suppose that X has the standard exponential distribution, so
FX (x) = P(X ≤ x) = 1 − e −x , x ≥ 0.
Now suppose that W = βX 1/α for some α > 0 and β > 0. Then
FW (w ) = 0 for w < 0, and for w ≥ 0
α w
1/α
FW (w ) = P(W ≤ w ) = P(βX
≤ w) = P X ≤
β
α
= 1 − e −(w /β) .
W has the Weibull distribution with parameters α and β.
1 / 10
Continuous Random Variables
Other Distributions
ST 380
Probability and Statistics for the Physical Sciences
The pdf of the Weibull distribution is found by differentiating FW (w ).
Like the gamma family, the Weibull family includes the exponential
distribution as a special case, here when α = 1.
Weibull distributions have been used as models for failure times in
survival analysis and reliability theory and for various physical
measurements.
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Continuous Random Variables
Other Distributions
ST 380
Probability and Statistics for the Physical Sciences
Lognormal Distribution
Suppose that X ∼ N(µ, σ 2 ), so that
P(X ≤ x) = Φ
x −µ
σ
.
Now suppose that Y = e X . Then FY (y ) = 0 for y ≤ 0, and for y > 0
log y − µ
X
FY (y ) = P(Y ≤ y ) = P(e ≤ y ) = P(X ≤ log y ) = Φ
.
σ
Y has the lognormal distribution with parameters µ and σ.
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Continuous Random Variables
Other Distributions
ST 380
Probability and Statistics for the Physical Sciences
The pdf of the lognormal distribution is found by differentiating
FY (y ).
The pdfs in the lognormal family are less varied in shape than those
of the gamma and Weibull families, but have been found useful as
models for particle size distributions and air pollution levels.
They also play a prominent role in the Black-Scholes theory of the
prices of financial options such as puts and calls.
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Continuous Random Variables
Other Distributions
ST 380
Probability and Statistics for the Physical Sciences
Beta Distribution
The gamma, Weibull, and lognormal families are models for
non-negative and unbounded continuous random variables.
The beta family is a model for a bounded continuous random variable.
In the simplest case, the range of X is (0, 1), and for parameters
α > 0 and β > 0, the pdf is
(
Γ(α+β) α−1
x (1 − x)β−1 0 < x < 1
Γ(α)Γ(β)
f (x; α, β) =
0
otherwise.
If the variable should have a more general range (A, B), use the
variable Y = A + (B − A)X .
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Continuous Random Variables
Other Distributions
ST 380
Probability and Statistics for the Physical Sciences
The beta family includes the uniform distribution as the special case
α = β = 1.
Wigner’s semi-circle density is another special case, with α = β = 1.5
and A = −B:
( √
2
B 2 − x 2 |x| < B
2
f (x; 1.5, 1.5, −B, B) = πB
0
|x| ≥ B.
Applications of the beta family include:
the distribution of grades on an exam (A = 0, B = 100);
amounts recovered in a bankruptcy (if in cents per dollar,
A = 0, B = 100; if as a fraction, A = 0, B = 1).
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Continuous Random Variables
Other Distributions
ST 380
Probability and Statistics for the Physical Sciences
Probability Plots
Many statistical procedures depend on assumptions about the nature
of the data being analyzed, such as that they are normally distributed.
So we need ways to explore whether such assumptions are correct.
The quantile-quantile plot (or q-q plot) is a graphical tool for doing
that.
7 / 10
Continuous Random Variables
Probability Plots
ST 380
Probability and Statistics for the Physical Sciences
The basic idea is to plot the quantiles (percentiles) of the distribution
of one random variable against those of another.
If they are all the same, the graph is the identity line y = x, and
the distributions are identical.
If the graph is some other straight line, the distributions are in
the same location-scale family.
Suppose for example that X ∼ N(0, 1) and Y ∼ N(µ, σ 2 ). We have
seen that
ηY (p) = µ + σηX (p),
so the graph of ηY (p) against ηX (p) is the straight line y = µ + σx.
8 / 10
Continuous Random Variables
Probability Plots
ST 380
Probability and Statistics for the Physical Sciences
In practice, we may know (or hypothesize) the distribution of X , for
example as the standard normal distribution, but have only a sample
of values from the distribution of Y .
So we estimate the quantiles of the distribution of Y , and plot the
estimates of those quantiles against those of X .
For instance, the median of the sample values is an estimate of the
median of the distribution of Y , the sample quartiles estimate the
quartiles of the distribution of Y , and so on.
9 / 10
Continuous Random Variables
Probability Plots
ST 380
Probability and Statistics for the Physical Sciences
More generally, if the ordered sample values are
.
y(1) ≤ y(2) ≤ · · · ≤ y(n) , then y(i) estimates ηY i−1/2
n
A q-q plot is a graph of
i − 1/2
i − 1/2
= y(i) against ηX
, i = 1, 2, . . . , n.
η̂Y
n
n
The R function qqnorm() actually plots
i −a
y(i) against ηX
,
n + 1 − 2a
where a = 3/8 for n ≤ 10 and a = 1/2 otherwise.
10 / 10
Continuous Random Variables
Probability Plots