1.1 Basic Concepts and Formulae
11
Two limiting cases:
(a) t 2 = ∞; N = N o e
−λt
(Law of radioactivity)
(1.72)
This gives the number of surviving atoms at time t.
(b) t 1 = 0; N = N o (1 − e
−λt )
(1.73)
For radioactive decays this gives the number of decays in time interval 0 and t.
Above formulas are equally valid for length intervals such as interaction lengths.
Moment generating function (MGF)
MGF = Ee
(x−μ)t
= E
1 + (x − μ)t + (x − μ)
2 t
2
2!
+ . . .
= 1 + 0 + μ 2
t
2
2!
+ μ 3
t
3
3!
+ . . .
(1.74)
so that μ n , the nth moment about the mean is the coefficient of t
n
/n!.
Propagation of errors
If the error on the measurement of f (x, y, . . .) is σ f and that on x and y, σ x and σ y ,
respectively, and σ x and σ y are uncorrelated then
σ
2
f =
∂ f
∂ x
2
σ
2
x +
∂ f
∂ y
2
σ
2
y + · · ·
(1.75)
Thus, if f = x ± y, then σ f =
σ
2
x + σ
2
y
1/2
And if f =
x
y
then
σ f
f
=
σ
2
x
x 2 +
σ
2
y
y 2
1/2
Least square fit
(a) Straight line: y = mx + c
It is desired to fit pairs of points (x 1 , y 1 ), (x 2 , y 2 ), . . . , (x n , y n ) by a straight line
Residue: S =
n
i=1 (y i − mx i − C)
2
Minimize the residue:
∂s
∂m
= 0;
∂s
∂c
= 0
The normal equations are:
m
n
i=1
x
2
i + C
n
i=1
x i −
n
i=1
x i y i = 0
m
n
i=1
x i + nC −
n
i=1
y i = 0