Covariance Matrix

Reference for Covariance and Covariance Matrices from wiki.

Variance : measure how much a single random variable varies
Covariance: measure how much two random variable varies together
Covariance Matrix: covariance in the form of matrix for higher dimension of scalar-valued random variable

Refernce for matrix from wiki

Continuous Distribution

EXPONENTIAL

Used to model the amount of time until a specific event occurs or to model the time between the independent events. Example:

• the time until the computer locks up
• the time between arrivals of telephone calls
• the time until a part fails

MATLAB: expocdf(x,1/λ) where mean E(X)=1/λ.

GAMMA f(x;λ,t)

when t is a positive integer, the gamma distribution can be used to model the amount of time one has to waith until the t events has occurred.

MATLAB: gampdf(x,t,1/λ).

CHI-SQUARE

A gamma distribution where λ=0.5 and t= v/2 where v is a positive integer, is called a chi-square distribution with v degree of freedom. Chi-square distribution is used to derived the distribution of the sample variance and is important for goodness-fit-test in statistical analysis.

WEIBULL

Closely related to Exponential.
Apply for problems of reliability and life testing.
Used to model the distribution of time it takesa for objects to fail.

BETA

Can be used to model ran var that takes on value over a bounded interval from 0 to 1.

MULTIVARIATE NORMAL

Normal Distribution [Miller 1985 n Martinez 2001]

History:

1. Known as Gaussian Distribution.
2. Studied first in 18th century when scientists observed an astonishing degree of regularity in errors of measurement.
3. The error distributions observed were approximated by distribution called ' normal curve of errors' (Bell shape) produced by the normal distrbution Eqn. that determined by the expected value and variance for normal distribution.

Properties:

1. PDF aproaches zero as x approaches + or - inf
2. centered at mean μ and max value occur at x=μ
3. PDF for normal distribution is symmetric about mean μ

MATLAB Command:

1. normcdf(x,mu,sigma)
2. normpdf(x.mu,sigma)
3. normspec(specs, mu, sigma)

MATLAB Example:
%Set up parameter for normal distb.
mu = 5;
sigma = 2;
%Set up upper and lower limit specs
specs = [2, 8]prob = normspec(specs, mu, sigma);

Equations:

GAUSSIAN ( NORMAL ) DISTRIBUTION ( PDF ), MEAN, AND VARIANCE


STANDARDIZED RAN VAR, STANDARD NORMAL DISTRIBUTION ( CDF )

Definition: Standard Nornal Distribution is a normal probability distribution that has a mean of 0 and a standard deviation of 1

GAUSSIAN APPROXIMATION TO THE BINOMIAL DISTRIBUTION

Use to approximate the binomial distribution when n is large but but is close to 0.5, not small enough to use Poisson Approximation.
Rule of thumb: use the normal approximation to the binomial distribution only when np and (1-np) are both greater than 5.

Theorem: (State without proof) If x is a value of random variable having the binomial distribution with the parameters n and p and if
then the limiting form of the distribution function of this standardized random variables as n --> inf is given by

Uniform Distribution [Miller 1985] MATLAB [Martinez 2001]

Uniform distriobution for Continuous random variables. Random variables are uniformly distributed over the interval (a,b).



















EXAMPLE MATLAB

Poisson distribution [Miller n Freund] Matlab [Martinez]

Poisson Distribution is approximation for Binomial Distribution when n --> inf and p --> 0, smalll ( so np is moderate).
Derived from Binomial Dist Eqn by sub var p with λ/n .

where λ= np
Expected value E[X] = λ and variance V(X) = λ (replace np=λ p-->0)

Example:

5% of of bounded book at certain bindery centre have defective. Find the probability that 2 of 100 books bounded by this bindery centre is defective:












Poisson Process:

Extending the uses of above formula for process taking place over continuous interval of time. i.e: Events occur at points in time or space

To find the probability of x success during a time interval of length T, we devided the interval T into n equal parts of length ∆t, with the probability of success p = α ∆t. α is the average (mean) of successes per unit time.

Assumption:

1. The probability of a success during a very small interval, ∆t, is given by p = α ∆t.
2. The probability of > one success during such a small time interval ∆t is negligible.
3. The probability of a success during such a time interval does not depend on what happened prior to that time.
   

The formula for Poisson distribution can be futher expand by expading the parameter λ.
λ = n.p = (T/∆t) *(α∆t) = αT

Note: However most of the time we use symbol λ to represent α .

Example:

Bank receives average λ= 6 bad checks per day, what are the probabilty that it will receive:
a: 4 bad checks on any given day.

• f(x;λT) = f(4;6(1))=0.135
• MATLAB: prob = poisspdf(4,6) = 0.1339
• or prob = poisscdf(4,6)-poisscdf(3,6) = 0.1339
  

b: at most 10 bad checks on any two consecutive day.

• f(x;λT) = f(x;6(2))=f(0;12) + f(1; 12)+ ...+ f(10;12)= 0.347
• MATLAB: prob = sum(poisspdf(0:10,12) = 0.3472
• or prob = poisscdf(10,12) = 0.3472
  

Binomial Distribution [Miller and Freund] Matlab [ Martinez 2001]

Repeated trials with getting x successes in n trials i.e, x successes and n-x failures in n attempts.
Assumption involves:

1. Only 2 possible outcomes for each trial: success and failure
2. The probability of success is the same for each trial
3. There are n trials, where n is constant
4. The n trial are independent

Trials satisfying this assumptions are reffered to as Bernoulli random variables.

for x = 0, 1, 2, ... , n

Expected Value (mean) E[X]= np ; Variance V(x) = np(1-p)
discriptions:

n : trials
x : successes
n - x : failures
p : probability of success
1-p : probability of failure
| n|
| x| : called binomial coefficient , which is the no of combination of x objects selected from a set of n object = n! / (r!(n-r)!)

MATLAB EXAMPLE on Binomial distribution using both probability mass function and cummulative distribution function.