FEC FINANCIAL ENGINEERING CLUB

WELCOME!

Facebook: http://www.facebook.com/UIUCFEC

LinkedIn: http://www.linkedin.com/financialengineeringclub

Email: uiuc.fec@gmail.com Please Welcome the MSFE

Director, Morton Lane!

fecuiuc.comis up!

PROBABILITY & STATISTICS PRIMER

DISCRETE RANDOM VARIABLES

Definition: The cumulative distribution function (CDF), of a random variable X is defined by

Definition: A discrete random variable, X, has probability mass function (PMF) if and for all events we have

Definition: The expected value of a function of a discrete random variable X is given by

Definition: The variance of any random variable, X, is defined as

BERNOULLI & BINOMIAL RVS

Bernoulli RV: Let X=Bernoulli(p) Pdf:

Binomial RV:

PDF:

Models: The probability that we achieve successes after trials, each with probability of

success

POISSON RVS

Let

Models: The probability that some event occurs times in a fixed time period if

the event is known to occur at an average rate of times per time period, independently of the last event.

GEOMETRIC DISTRIBUTION

Let

Models: The probability that it takes successive independent trials to get first

success with probability of success for each event

CONTINUOUS RANDOM VARIABLES

Definition: A continuous random variable, X, has probability density function (PDF) if and for all events we have

Definition: The cumulative distribution function (CDF), of a continuous random variable X is related to the PDF by:

Definition: The expected value of a function of a continuous random variable X is given by

EXPONENTIAL

Let

PDF:

Models: The time between events occurring independently and continuously at a constant average rate

NORMAL/GAUSSIAN DISTRIBUTION

Let

Central Limit Theorem:

Let be a sequence of

independent random variables with mean

and variance . Then:

BROWNIAN MOTION

BROWNIAN MOTION

0 100 200 300 400 500 600-20

0

20

40

60

80

100

120

u=1 var=100

u=3 var=800

u=1 var=300

SIMULATING RANDOM VARIABLES

For continuous, use inverse CDF method: if F(x) is cdf of random variable X then to simulate X, Generate U~Uniform(0,1) X = Easy example: simulate an exponential with parameter λ

CDF if x ≥ Simulate U~Uniform(0,1), note that (1-U)~Uniform(0,1) Set X = , X is exponential(λ)

CONDITIONAL PROBABILITY

Definition: The probability that X occurs given Y occurred is:

Bayes’s Theorem says that:

MULTIVARIATE RANDOM VARIABLES

We have two RVs, X and Y

Let the joint PDF of X and Y be

Definition: The joint cumulative distribution function (CDF) of satisfies

Definition: The marginal density function of is:

MULTIVARIATE RANDOM VARIABLES

MULTIVARIATE RANDOM VARIABLES

INDEPENDENT RANDOM VARIABLES

INDEPENDENT RANDOM VARIABLES

COVARIANCE

Covariance is the measure of how much two variables change together. Cov(X,Y)>0 if increasing X increasing Y Cov(X,Y)<0 if increasing X decreasing Y

CORRELATION COEFFICIENT

Definition: The correlation of two RVs, X and Y, is defined by:

If X and Y are independent, they are uncorrelated:

VARIANCE AND COVARIANCE

VARIANCE AND COVARIANCE

LINEAR REGRESSION

Least Squares Method:

The minimizing is:

The minimizing is:

COMBINATIONS OF RANDOM VARIABLES

Examples, portfolio mean and variance: Equations (1) and (3) generalized to N variables (assets in the portfolio) with coefficients as weights: see boxed info in http://en.wikipedia.org/wiki/Modern_portfolio_theory

MOMENT GENERATING FUNCTIONS

…..

GEOMETRIC BROWNIAN MOTION

MAXIMUM LIKELIHOOD ESTIMATOR

Likelihood function = Let represent all parameters to the RV is a function of , fixed

is the maximum likelihood estimator (MLE)

THANK YOU!

Facebook: http://www.facebook.com/UIUCFEC

LinkedIn: http://www.linkedin.com/financialengineeringclub

Email: uiuc.fec@gmail.com Next Meeting:“Trading and

Market Microstructure”

Wed. 26th Feb. 6-7pm

165 Everitt

fecuiuc.comis up!