Transcript Summary Statistics When analysing practical sets of data

Standard Statistical Distributions
Most elementary statistical books provide a survey of commonly used statistical
distributions. The reason we study these distributions are that
They provide a comprehensive range of distributions for modelling practical applications
Their mathematical properties are known
They are described in terms of a few parameters, which have natural interpretations.
1 Prob
1. Bernoulli Distribution.
This is used to model a trial which gives rise to two outcomes:
success/ failure, male/ female, 0 / 1. Let p be the probability that
the outcome is one and q = 1 - p that the outcome is zero.
1-p
E[X]
= p (1) + (1 - p) (0) = p
VAR[X] = p (1)2 + (1 - p) (0)2 - E[X]2 = p (1 - p).
0
2. Binomial Distribution.
Suppose that we are interested in the number of successes X
in n independent repetions of a Bernoulli trial, where the
probability of success in an individual trial is p. Then
Prob{X = k} = nCk pk (1-p)n - k, (k = 0, 1, …, n)
E[X]
=np
VAR[X] = n p (1 - p).
This is the appropriate distribution to use in modeling the
number of boys in a family of n = 4 children, the number of
defective components in a batch n = 10 components and so on.
Prob
p
p
1
(n=4, p=0.2)
1
np
4
3. Poisson Distribution.
The Poisson distribution arises as a limiting case of the binomial distribution, where
n , p  in such a way that n p  a constant). Its density is
Prob{X = k} = exp ( - )k /k!k=,1,2,… ).
Note that exp (x) stands for e to the power of x, where e is
Prob
approximately 2.71828.
1
E [X]
=
VAR [X] = .
The Poisson distribution is used to model the number of
occurrences of a certain phenomenon in a fixed period of
time or space, as in the number of
5
O particles emitted by a radioactive source in a fixed direction and period of time
O telephone calls received at a switchboard during a given period
O defects in a fixed length of cloth or paper
O people arriving in a queue in a fixed interval of time
O accidents that occur on a fixed stretch of road in a specified time interval.
X
4. Geometric Distribution.
This arises in the “time” or number of steps k to the first
success in a series of independent Bernoulli trials. The
density is
Prob{X = k} = p (1 - p) k-1 (k = 1, 2, … ).
E[X] = 1/p
VAR [X] = (1 - p) /p2
Prob
1
X
5. Negative Binomial Distribution
This is used to model the number of failures k that occur before the rth success in a series of
independent Bernoulli trials. The density is
Prob {X = k} = r+k-1Ck pr (1 - p)k
(k = 0, 1, 2, … )
Note
E [X]
= r (1 - p) / p
VAR[X]
= r (1 - p) / p2.
6. Hypergeometric Distribution
Consider a population of M items, of which W are deemed to be successes. Let X be the
number of successes that occur in a sample of size n, drawn without replacement from the
population. The density is
Prob { X = k} = WCk M-WCn-k / MCn ( k = 0, 1, 2, … )
Then
E [X] = n W / M
VAR [X] = n W (M - W) (M - n) / { M2 (M - 1)}
7. Uniform Distribution
Prob
1
A random variable X has a uniform distribution on
the interval [a, b], if X has density
f (X) = 1 / ( b - a)
for a < X < b
1 / (b-a)
X
=0
otherwise.
Then
E [X] = (a + b) / 2
a
b
VAR [X] = (b - a)2 / 12
Uniformly distributed random numbers occur frequently in simulation models. However,
computer based algorithms, such as linear congruential functions, can only approximate this
distribution so great care is needed in interpreting the output of simulation models.
If X is a continuous random variable, then the probability that X takes a value in the range
[a, b] is the area under the frequency function f(x) between these points:
Prob { a < x < b } = F (b) - F (a) = ab f(x) dx.
In practical work, these integrals are evaluated by looking up entries in statistical tables.
9. Gaussian or Normal Distribution
A random variable X has a normal distribution with mean m and standard deviation s if it
has density
f (x)
=
1
Prob
1
exp { - ( x - m )2 }, -x < 
2ps22s2
=
0,
otherwise
f(x)
E [ X]
=m
VAR [X] = s2.
0
m
As described below, the normal distribution arises naturally as the limiting distribution of the
average of a set of independent, identically distributed random variables with finite
variances. It plays a central role in sampling theory and is a good approximation to a large
class of empirical distributions. For this reason, a default assumption in many empirical
studies is that the distribution of each observation is approximately normal. Therefore,
statistical tables of the normal distribution are of great importance in analysing practical data
sets. X is said to be a standardised normal variable if m = 0 and s = 1.
X
10. Gamma Distribution
The Gamma distribution arises in queueing theory as the time to the arrival of the n th
customer in a single server queue, where the average arrival rate is . The frequency
function is
f(x)
=  ( x )n - 1 exp ( -  x) / ( n - 1)! , x  0,  >0, n = 1, 2, ...
= 0,
otherwise
E [X]
=n/
VAR [X] = n /  2
11. Exponential Distribution
This is a special case of the Gamma distribution with n = 1 and so is used to model the
interarrival time of customers, or the time to the arrival of the first customer, in a simple
queue. The frequency function is
f (x)
=  exp ( -  x ),
x 0,>0
= 0,
otherwise.
12. Chi-Square Distribution
A random variable X has a Chi-square distribution with n degrees of freedom ( where n is a
positive integer) if it is a Gamma distribution with  = 1, so its frequency function is
f (x)
= xn - 1 exp ( - x) / ( n - 1) !, x o
Prob
= 0,
otherwise.
c2 n (x)
X
Chi-square Distribution (continued)
The chi-square distribution arises in two important applications:
O If X1, X2, … , Xn is a sequence of independently distributed standardised normal
random variables, then the sum of squares X12 + X22 + … + Xn2 has a chi-square
distribution with n degrees of freedom
O If x1, x2, … , xn is a random sample from a normal distribution with mean m
and variance s2 and let
x =  xi / n
and S2 =  ( xi - x ) 2 / s2,
then S2 has a chi-square distribution with n - 1 degrees of freedom, and the
random variables S2 and x are independent.
13. Beta Distribution.
A random variable X has a Beta distribution with parameters a>0 and b>0 if it has
frequency function
f (x)
= Ga+b) x a- 1 ( 1 - x) b- 1 /G (a) Gb), 0 < x < 1
= 0,
otherwise
E [X]
= a/a+b)
VAR [X] = ab/[a+b)2 a+b+1)]
If n is an integer,
G (n) = ( n - 1 ) !
G (n + 1/2) = (n - 1/2) ( n - 3/2) …
with G (1) = 1
with G ( 1/2) =  p
14. Student’s t Distribution
A random varuable X has a t distribution with n degrees of freedom ( tn ) if it has density
f(x)
= G (n+1) / 2 )
n pGn / 2)
1 + x2 / n ) - (n+1) / 2
= 0,
The t distribution is symmetrical about the origin, with
E[X]
=0
VAR [X] = n / (n -2).
( - < x < )
otherwise.
For small values of n, the tn distribution is very flat. As n is increased the density assumes a
bell shape. For values of n  25, the tn distribution is practically indistinguishable from the
standard normal curve.
O If X and Y are independent random variables
If X has a standard normal distribution and Y has a cn2 distribution
then
X
has a tn distribution
Y/n
O If x1, x2, … , xn is a random sample from a normal distribution, with mean m
and variance s2 and if we define s2 = 1 / ( n - 1)  ( xi - x ) 2
then ( x - m ) / ( s / n) has a tn- 1 distribution
15. F Distribution
A random variable X has an F distribution with m and n degrees of freedom if it has density
f(x)
= G (m + n) / 2 ) m m / 2 n n / 2 x m / 2 - 1
Gm / 2) Gn / 2) (n + m
Note
If
x>0
x) ( m + n ) / 2
= 0,
E[X]
= n / ( n - 2)
VAR [X] = 2 n2 (m + n - 2)
m (n - 4) ( n - 2 )2
otherwise.
if n > 4
if n > 4
O X andYare independent random variables, X has a cm2 and Y a cn2 distribution
X / m has an Fm , n distribution
Y/n
O One consequence of this is that the F distribution represents the distribution of
the ratio of certain independent quadratic forms which can be constructed from
random samples drawn from normal distributions:
if x1, x2, … , xm ( m 2) is a random sample from a normal
distribution with mean m1 and variance s12, and
if y1, y2, … , yn ( n 2) is a random sample from a normal
distribution with mean m2 and variance s22, then
 ( xi - x )2 / ( m - 1)
 ( yi - y )2 / ( n - 1)
has an Fm - 1 , n - 1 distribution