Objectives:
To study the situations where gamma distribution is suitable. To study the relation between exponential and gamma distribution, normal and gamma distribution.
Introduction
We have discussed use of exponential distribution as a life time distribution. Another distribution popularly applied when variable takes non-negative values is Gamma distribution. It can be looked upon as a sum of i.i.d. exponential random variables. Just like exponential distribution it plays vital role in the theory of queues, reliability and survival analysis. It belongs to the group of extremal distributions, since it is used to model the distributions extreme values such as maximum or minimum rainfall, wind velocity, humidity etc. It is used to model the size of insurance claims.
Required Results: Gamma Integral (or Function)
The integral ex xdx is called as a gamma function or gamma
integral. It is denoted by n when n is a positive real number. There are two more forms of gamma integrals as follows:
n
e-xx-1 dx = "a"
when a and n are positive real numbers.
b/2
e-ax2xb-1 dx =
a>0, b>0, a and b € R.
0
Note the properties of [n:
(i) n = (n-1)n-1.
(ii) If n is integer then n = (n-1)!
(iii) = π
(iv) Beta function (or integral) B(m, n) = x (1 - x)" dx m,
mn
n>0 then B(m, n) =
(m + n)
0
P.D.F. of Gamma Distribution
A continuous random variable X taking non-negative real values is said to follow gamma distribution with parameters α and λ if its p.d.f. is,
f(x) =
αλ Γ (λ)
e-ax x2-1 ; x≥0, α, λ>0
= 0
otherwise.
we write it as X → G (α, λ)
Remarks:
(1) We shall verify that f (x) is a p.d.f.
(i) clearly f (x) > 0 for x≥0, a > 0, 2>0
(i) f(x) dx = 0
=
αλ
Γ (λ)
0
αλ r (2)
e-ax xλ-1 dx
e-ax xλ-1 dx
}
e-axx-1 dx =
Γ() = 1
Γ (λ) αλ
Here, nλ, a = α
(2) In particular when λ= 1, the p.d.f. of X is given by, f(x) α eax; x>0
which is p.d.f. of exponential distribution with mean (a), when λ= 1.
1
i.e. X -> Exp
α
(3) When both parameters α and λ are equal to one then p.d.f. of X is, f (x) = ex; x≥0
Thus, in this case X follows standard exponential distribution.
(4) Nature of probability density curve is as shown in Fig. 1.1. Gamma distribution probability curve
Alpha-1,landa=0.5
5
Alpha 1 lamda >
alpha=1,lamdu=3
Fig. 1.1
Mean and variance of G (α, λ)
Mean = E(X)
x f (x) dx
0
αλ
X
e-ax xλ-1 dx
r (2)
Γ (λ)
αλ
Γ (λ)
e-ax xλ dx
· 0
e-ax x(2+1) - 1 dx
αλ
=
Γ (λ)
Γ ( + 1) αλ + 1
Here n = λ+
a = α
|E(X) ==
[. Γ (+1) = λ Γ (λ)]
Variance: To obtain variance let us find E (X2)
..
E (X) =
x2 f (x) dx
0
αλ
= x2
e-xx xλ - 1 dx
Γ (λ)
0
αλ
=
e-ax xλ +1 dx
г (2)
=
50
e-ax x(+2)-1dx
=
αλ
Γ (λ)
αλ г(2)
αλ
=
г (2)
Γ (λ + 2) α+2
(λ + 1) Γ () (2+1) 2
αλ +2
Var (X) = E(X2)-E (X)2
=
(2+1) α
x2+λ-22
=
Q
a2
..
Var (X) =
Moments of G (α, λ)
First we shall obtain the expression for rth raw moment, in terms of
α and λ.
= 0
αλ
xr
.e-ax xλ-1 dx
Γ (λ)
H = E(X) =
=
αλ
Γ (λ)
0
0
e-ax x(+r) -1 dx
are,
αλ
Γ ( + )
=
г (2)
αλ+r
a
H
=
αλτι
=
+r-1) (λ + r− 2) ... λ г (2) Γ (λ)
λ (λ + 1) ... ( + r − 2) (λ + r − 1)
απ
... (1)
From (1) putting r = 1, 2, 3, 4 first four raw moments of G (a, λ)
μ, = λα
μ2 = λ(2+1)/α2
μ. = (λ + 1) (λ + 2) / α
μ1 = 2 (2+1) (λ + 2) (λ + 3) / α+
The first four central moments can be obtained using usual relations among the central and raw moments as,
μ1 = 0
μ2 = variance = μ2-(u)2
λ(2+1) 22
= απ
α
4. -3м, ні +2 (4)
μ. = μ
2 + 1) (λ + 2)
α
- 3 [^ (^ + ']) + 2)2
22 α
Note: μ is positive since a and 2 are positive
+6μ (μ)2 - 3 (μ)+
^ @λ + 1) @ + 2) Ɑ + 3) _ 4 (^ @ + 1)(α + 2)) (~)
04
α
α
++-A
[~ + 1) (A + 2) (A + 3) − 42 (2 + 1) (2 + 2) + 622 (^x + 1) − 323]
32 (2+2)
=
(32+6) =
Q4
3 λ (λ + 2) 04
Coefficients of skewness and kurtosis
με 422 α
4
B1 ==
X
=
H2
α
入
Y1 =
VB. = +
唁
[sign of y, is positive because μ, is positive]
Hence, the distribution is positively skew.
μ4 3λ(2+2)
B2 = 2 =
=
με
04
..
B2 =
32+6 λ
6
3+ >3
λ
6
..
Y1⁄2 = B2-3=> 0
Thus, the distribution is leptokurtic.
Note:
4
(1) If λ→ ∞, ẞ, =
→0
. For large λ, gamma distribution
becomes symmetric.
As 24, B,
1.
6
(2) If λ→ ∞, B2 =3+
→3. For large λ, gamma distribution becomes mesokurtic.
Moment Generating Function and C.G.F.
(A) M.G.F. of G (α, λ)
Mx (t) = E [etx] = []
=
0
0
ай
etx
I (2)
αλ Γ()
etx f (x) dx
e-ax xλ-1 dx
e-a-1)x x2-1, dx a-t>0 ort<α
0
αλ
Γ (λ)
αλ
г (λ) '(α-t)λ
(α-t)λ
αι -λ
c.g.f.
=
= (1-4):
Mx (t) =
<α or <1
; t <a
For a gamma distribution, it is easier to find central moments from
(B) Cumulant Generating Function (c.g.f) of G (α, λ)
Kx (t) = loge Mx (t)
30 (1 - ~ ) ^ -
= loge
=-2 loge
(1-4)
t/ar
+
4
r
+...]
t2/02 13/03 t4/04 +++
= [+
Additive Property of G (α, λ)
Statement: Suppose X, → G (α, λ) and X2 → G (α, λ) then
X1 + X2→G (α, λ, +22) provided X, and X2 are independent.
Generalisation of additive property statement:
If X1, X2 ... Xi, ... Xn are independent variables such that X; → G (α, λ;), i = 1, 2, ... n then X, +X2+ ... + Xn→ G (α, 21 +22+...+21)
n
Note:
(1) If X, → G (α, 2), X2 → G (α, 2) and X, and X2 are independent then X, - X2 does not follow gamma distribution.
(2) The additive property holds good for Gamma variates, only if
they have same value of parameter α.
G(a,λ)
(3) If XG (a, 2) then cX → G where c is a positive constant,