Upload
aliza
View
34
Download
0
Embed Size (px)
DESCRIPTION
FEC Financial Engineering Club. Welcome!. Facebook: http://www.facebook.com/UIUCFEC LinkedIn: http://www.linkedin.com/financialengineeringclub Email: [email protected]. Please Welcome the MSFE Director, Morton Lane!. f ecuiuc.com i s up!. Probability & Statistics Primer. - PowerPoint PPT Presentation
Citation preview
FEC FINANCIAL ENGINEERING CLUB
WELCOME!
Facebook: http://www.facebook.com/UIUCFEC
LinkedIn: http://www.linkedin.com/financialengineeringclub
Email: [email protected] 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: [email protected] Next Meeting:“Trading and
Market Microstructure”
Wed. 26th Feb. 6-7pm
165 Everitt
fecuiuc.comis up!