Distribution Basics
Statistics
• Descriptive statistics – Those methods involving the collection, summarisation, presentation
and characterisation of a set of data in order to properly describe the various features of that
set of data.
• Inferential statistics – Those methods that make possible the estimation of a characteristic of
a population or the making of a decision concerning a population based only on a sample.
Population vs. sample
• A population is a total collection of observations or measurements of interest
• A sample is a subset of measurements or observations from a population
Sample vs. population
m = Population mean
s = Population standard deviation
EstimateSamplestatistics
Populationparameters
SAMPLE POPULATION
s = Sample standard deviation (s)̂
X = Sample mean (m)̂
Describing the sample
• Central tendency – a distinct tendency to cluster about a central point (an average)
• Dispersion – amount of variation or spread in the data
• Shape – manner in which data are distributed
Statistical functions
• f(x) – Probability Density Function (PDF)
• Models a histogram
• F(x) – Cumulative Distribution Function (CDF)
• The area under f(x) to the left of x
• Probability of less than: integral of f(x) from negative infinity to x
• R(x) – Reliability function
• The area under f(x) to the right of x
• 1-F(x)
• Probability of greater than: integral of f(x) from x to infinity
• h(x) – Hazard function
• Instantaneous failure rate: f(x)/R(x)
• A measure of proneness to fail
Probability Density Function Definition
Symmetric vs. skewed data
• Mean > Median: Positive or right-skewness
• Mean = Median: Symmetry or zero-skewness
• Mean < Median: Negative or left-skewness
positively skewed (right) negatively skewed (left)
symmetric
Moment Generating Functions
1st moment generating function about the origin
2nd moment generating function about the mean
Skewness = the 3rd moment generating function about the origin
Example• Given the density function f(x) = ax, 0 < x < 10
• Determine the value of a that makes f(x) a valid density function
• Determine the mean and variance of the distribution
• Derive expressions for the reliability and hazard functions
( ) ( )( )ax dxa
xa
a0
102
0
10
2 21
21
210 0 1
1
50 = = − = =
( )E x xx
dxx
dx( ) .= = = = 50 50
1
15010 6 667
0
102
3
0
10
( ) ( )V x x dx( ) . . .= − = − =1
506 667
1
20010 44 449 5 55
3 2 4
0
10
Example Continued
Probability Density Function (PDF)
0.59
0.71
0.83
0.95
1.07
1.19
1.31
1.43
1.55
1.67
1.79
1.91
2.03
2.15
2.27
2.39
2.51
2.63
0
10
20
30
40
Count
Probability Density Function
0
x
f(x)
a b
6 8Diameter
4 6 80
0.2
0.4
0.6
0.8
1
Diameter
Cu
mu
lati
ve D
istr
ibu
tio
n
40
0.002
0.004
0.006
0.008
Pro
ba
bil
ity
De
nsi
ty
Cumulative Distribution Function (CDF)
Reliability Function
0x
1
F(x) R(x)
Hazard Function
• Instantaneous failure rate
