Applied Statistics Using SPSS, STATISTICA, MATLAB and R

Properties:
1. w1,1/λ(x) ≡ ελ(x).
2. w2,1/λ (x) is the so-called Rayleigh distribution.
3. X ~ ε λ ⇒
α
X ~ w
α ,
α .
1/
λ
4. X ~ w α ⇒
α
X ~ ε λ .
Example B. 11
α , 1/
λ
B.2 Continuous Distributions 445
Q: Consider that the time in years that an implanted prosthesis survives without
needing replacement follows a Weibull distribution with parameters α = 2, β =10.
What is the expected percentage of patients needing a replacement of the prosthesis
after 6 years?
A: P = W ( 6)
= 30.2%.
0.
5,
1
B.2.5 Gamma Distribution
Description: The Gamma distribution is a sort of generalisation of the exponential
distribution, since the sum of independent random variables, each with the
exponential distribution, follows the Gamma distribution. Several continuous
distributions can be regarded as a generalisation of the Gamma distribution.
Sample space: ℜ + .
Density function:
1 −x
/ a p−1
γ a,
p ( x)
= e x ,
p
a Γ(
p)
a,
p > 0 (0, otherwise), B. 22
with Γ(p), the gamma function, defined as ∫ ∞ −x
p−1
Γ(
p)
= e x dx , constituting a
0
generalization of the notion of factorial, since Γ(1)=1 and Γ(p) = (p − 1) Γ(p − 1).
Thus, for integer p, one has: Γ(p) = (p − 1)!
Distribution function:
Γ
x
a, p ∫0
a,
p
( x)
= γ ( t)
dt . B. 23
Mean: µ = a p.
Variance: σ 2 = a 2 Properties:
p.
1. γa,1(x) ≡ ε1/a(x).
2. Let X1, X2,…, Xn be a set of n independent random variables, each with
exponential distribution and spread factor λ. Then, X = X1 + X2 +…+ Xn ~
γ1/λ,n.