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FEC FINANCIAL ENGINEERING CLUB WELCOME! ο Facebook: http://www.facebook.com/UIUCFEC ο LinkedIn: http://www.linkedin.com/financialengineeringclub ο Email: [email protected] fecuiuc.com is up! Please Welcome the MSFE Director, Morton Lane! PROBABILITY & STATISTICS PRIMER DISCRETE RANDOM VARIABLES ο Definition: The cumulative distribution function (CDF), of a random variable X is defined by πΉ π₯ = π(π β€ π₯) ο Definition: A discrete random variable, X, has probability mass function (PMF) if π π₯ β₯ 0 and for all events π΄ we have π πππ΄ = π₯ππ΄ π(π₯) ο Definition: The expected value of a function of a discrete random variable X is given by ΞΌ = πΈ π(π) = π π(π₯π )π(π₯π ) ο Definition: The variance of any random variable, X, is defined as π 2 = πππ π = πΈ π β πΈ π 2 = πΈ π 2 β πΈ[π]2 BERNOULLI & BINOMIAL RVS ο Bernoulli RV: ο Let X=Bernoulli(p) ο Pdf: ο π π π’ππππ π = π ο π ππππ = 1 β π ο 0<π<1 ο Binomial RV: ο ο ο ο ο π΅πππππππ(π, π) = ππ’π ππ π π΅πππππ’πππ(π) PDF: p(k) = ππ ππ 1 β π πβπ πΈ π = ππ πππ π = ππ(1 β π) Models: ο The probability that we achieve π successes after π trials, each with probability of success π POISSON RVS ο Let π = ππππ π ππ(π) ο ππ·πΉ: π(π) = π βπ ππ π! ο π = 0,1,2,3, β¦ ο π>0 ο πΈπ = π ο πππ π = π ο Models: ο The probability that some event occurs π times in a fixed time period if the event is known to occur at an average rate of π times per time period, independently of the last event. GEOMETRIC DISTRIBUTION ο Let π = ππππππ‘πππ π ο 0<π<1 ο π π =π = 1βπ πβ1 π ο π = 1,2,3,4 β¦ ο πΈπ = 1 π ο πππ π = 1βπ π2 ο Models: ο The probability that it takes π successive independent trials to get first success with probability of success for each event = π CONTINUOUS RANDOM VARIABLES ο Definition: A continuous random variable, X, has probability density function (PDF) if π π₯ β₯ 0 and for all events π΄ we have π πππ΄ = π΄ π π₯ ππ₯ ο Definition: The cumulative distribution function (CDF), of a continuous random variable X is related to the PDF by: πΉ π₯ =π πβ€π₯ = π₯ π ββ π₯ ππ₯ ο Definition: The expected value of a function of a continuous random variable X is given by ΞΌ = πΈ π(π) = β π(π₯) ββ π π₯ ππ₯ EXPONENTIAL ο Let π = ππ₯ππππππ‘πππ(π) ο PDF: ο π π₯ = ππ βππ₯ ο πΈπ = 1 π ο πππ π = 1 π2 ο Models: ο The time between events occurring independently and continuously at a constant average rate NORMAL/GAUSSIAN DISTRIBUTION ο Let π = ππππππ(π’, π 2 ) ο ππ₯(π₯) = 1 2π2 π β π₯βπ’ 2 2π2 ο Central Limit Theorem: Let π1 , β¦ , ππ be a sequence of π independent random variables with mean π and variance π 2 . Then: π π=1 ππ β ππππππ(ππ’, ππ 2 ) ππ π β β BROWNIAN MOTION BROWNIAN MOTION 120 u=1 var=100 100 u=3 var=800 80 u=1 var=300 60 40 20 0 -20 0 100 200 300 400 500 600 SIMULATING RANDOM VARIABLES ο For continuous, use inverse CDF method: if F(x) is cdf of random variable X then to simulate X, ο Generate U~Uniform(0,1) ο X = Fβ1 (π) ο Easy example: simulate an exponential with parameter Ξ» ο CDF πΉ π₯ = 1 β π βΞ»π₯ if x β₯ 0 ο πΉ β1 π¦ = β1 ln(1 Ξ» β π¦) ο Simulate U~Uniform(0,1), note that (1-U)~Uniform(0,1) 1 ο Set X = Ξ» ln(π), X is exponential(Ξ») CONDITIONAL PROBABILITY ο Definition: The probability that X occurs given Y occurred is: π· ππ = π πβ©π π π ο Bayesβs Theorem says that: π· ππΏ π· πΏπ = π·(πΏ) π·(π) = π· π· ππΏ ππΏ π· πΏ +π· π·(πΏ) π πΏβ² π·(πΏβ² ) MULTIVARIATE RANDOM VARIABLES ο We have two RVs, X and Y ο Let the joint PDF of X and Y be π(π₯, π¦) ο Definition: The joint cumulative distribution function (CDF) of π satisfies πΉπ₯,π¦ π₯, π¦ = π π β€ π₯, π β€ π¦ = π¦ π₯ π ββ ββ ο Definition: The marginal density function of π is: ππ₯ π₯ = β π ββ π₯, π¦ ππ¦ π₯, π¦ ππ₯ππ¦ MULTIVARIATE RANDOM VARIABLES MULTIVARIATE RANDOM VARIABLES INDEPENDENT RANDOM VARIABLES INDEPENDENT RANDOM VARIABLES COVARIANCE ο Covariance is the measure of how much two variables change together. ο Cov(X,Y)>0 if increasing X β increasing Y ο Cov(X,Y)<0 if increasing X β decreasing Y ο πΆππ£ π, π = 0 ππ π & π πππ πππππππππππ‘ ο πΆππ£ π, π = πππ = πΈ[ π β π’π₯ π β π’π ] = πΈ ππ β π’ππ’π ο πΆππ£ π, π = πππ π ο πππ π π=1 ππ = π π=1 = π π=1 πΆππ£(ππ, ππ) π π=1 πππ ππ + 2 π π=1 π π=π+1 πΆππ£(ππ, ππ) CORRELATION COEFFICIENT ο Definition: The correlation of two RVs, X and Y, is defined by: π π₯, π¦ = πΆππ£(π,π) πππ π πππ(π) ο If X and Y are independent, they are uncorrelated: π π₯, π¦ = 0 VARIANCE AND COVARIANCE VARIANCE AND COVARIANCE LINEAR REGRESSION ο Least Squares Method: π = πΆ + π·π π π=π(ππ βπΆ ππππππππ β π·ππ )π ο The minimizing π½ is: π½= πΆππ£(π₯,π¦) πππ(π₯) = ππ₯π¦ π π¦ π π₯ ο The minimizing πΌ is: πΌ = ππ£π π¦ β π½ β ππ£π(π₯) COMBINATIONS OF RANDOM VARIABLES ο πΏπππππππ‘π¦ ππ ππ₯ππππ‘ππ‘πππ: πΈ[ππ + π π] = ππΈ[π] + ππΈ[π] ο πΆππ£(ππ + ππ, ππ + ππ) = πππΆππ£(π, π) + πππΆππ£(π, π) + πππΆππ£(π, π) + πππΆππ£(π, π) ο πππ[ππ + ππ] = π2 πππ(π) + 2πππΆππ£(π, π) + π 2 πππ(π) ο 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 ο ππ₯ π‘ = πΈ π π₯π‘ ο ππ₯β² 0 = πΈ π ο ππ₯β²β² 0 = πΈ π 2 ο β¦.. ο ππ₯π 0 = πΈ[π π ] GEOMETRIC BROWNIAN MOTION MAXIMUM LIKELIHOOD ESTIMATOR ο Likelihood function πΏ(π|π) = π(π|π) ο Let π represent all parameters to the RV π ο πΏ is a function of π, π fixed ο πππ₯π (πΏ(π|π)) ο ππππ₯ is the maximum likelihood estimator (MLE) THANK YOU! ο Facebook: http://www.facebook.com/UIUCFEC ο LinkedIn: http://www.linkedin.com/financialengineeringclub ο Email: [email protected] fecuiuc.com is up! Next Meeting: βTrading and Market Microstructureβ Wed. 26th Feb. 6-7pm 165 Everitt
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