Moments of probability distributions
• The moments of a probability distribution are a
way of characterising its position and shape.
• Strong physical analogy with moments in
mechanics of rigid bodies
– Centre of gravity
– Moment of inertia
– Higher moments
Mean and median
• Mean value (centre of gravity)
f(x)
<x>
x   f (x) x dx
• Median value (50th percentile)
1
F(x med ) 
2
P(x  xmed )  P(x  x med )
x
F(x)
1
1/2
0
xmed
x
Variance and standard deviation
• Standard deviation  measures width of
distribution.
• Variance  (moment of inertia)
 2 = Var(x)  [x  x ]2
f(x)
<x>
  f (x) [x  x ]2 dx
-
+
x
Example: Gaussian distribution
G(,2)
• Also known as a normal distribution.
• Physical example: thermal Doppler broadening
1
f (x) 
e
2
1 x   2
 

2   

f(x)
• Mean value: <x> = 
• Variance:
x
-
• Full width at half maximum value (FWHM)
FWHM  8 ln 2  2.3
• 32% probability that a value lies outside  ±
• 4.5% probability a value lies outside  ±2
• 0.3% probability a value lies outside  ±3
 +
x
Higher central moments
• General form:
mn  x  x 
• e.g. Skewness (m3):
• e.g. Kurtosis (m4):
n
f(x)
x
f(x)
Peaky
Boxy
x
(Pathological) example: Lorentzian
(Cauchy) distribution
• Physical example: damping
wings of spectral lines.

1
f (x) 
  2  (x  )2
1
1 x   
F(x)  tan 

  

• Wings are so wide that no
moments converge!
x 
(x   )2  
f(x)
0.4
1.4
0.2
1.2
0
1
-4
-2
0
2
4
-0.2
x/
0.8
F(x)
-0.4
0.6
-0.6
0.4
-0.8
0.2
-1
0
-4
-2
0
2
4
x/
Poisson distribution P()
14
6
=5
4
2
19
17
15
13
11
9
7
0
5
•  = mean count rate
Bin number
f (x)  e
0.4
x



 (x  i)

x!
i0
0.3
P(x  n)  e  
0.2
0.1
11
7
9
x
5
3
0
1
P
8
1
– Raindrops in bucket per time interval
– Cars on road per time interval
– Photons per pixel during exposure
10
3
• A discrete distribution
• Describes counting statistics:
Counts per bin
12
8
S3
4
2
S1
1 
n
n!
n  0,1, 2,...
x 
 2 (x)     (x) 
x
Exponential distribution
• Distribution of time intervals between events
– Raindrops, cars, photons etc
f( x)
• A continuous distribution
1
0.5
0
0
1
2
3
2
3
x/ 
1
f (x)  e
1
x/
F(x)  1  e
F(x)

0.5
0
 x /
0
x    mean time between events
1
x/ 