Lecture 2

Lecture 2 - Probability distribution Outcomes: mutually...

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Unformatted text preview: Probability distribution Outcomes: mutually exclusive and collectively exhaustive Probability: relative frequency Cumulative probability distribution The probability that a random variable Y is less than or equal to a particular value y; Y for random variable and y for a number. Example: computer crashes when writing a term paper Bernoulli random variable: Regression with binary dependent variables Example: mortgage loan applications; credit card applications; default risk Continuous random variable Cumulative probability distribution Probability distribution Normal distribution: The probability distribution function of a normal random variable X is: = location parameter , = scale parameter (lower sigma indicates more spread out) We usually denote a normal random variable as X N(,). When =0 and =1, we have a standard normal distribution. We usually denote a standard normal random variable as z, with the cumulative distribution function being: P(Zz)=(z) If X N(,), then Y=(X-)/ is a standard normal . variable : Example X N(2,4), what is P(X<1.8)? P(X>2)? P(1.5<X<3)? mean = expected value (expectation) of Y = E ( Y ) = Y = long-run average value of Y over repeated realizations of Y variance = E ( Y Y )P 2 = 2 Y = measure of the squared spread of the Distribution standard deviation = variance = Y Example: Example: Example: If Y N(,), then the mean of Y is , the variance of Y is . Useful formulas: Random variables X and Z have a joint distribution Pr(X=x, Z=z) Marginal distribution E(M) = 0*.8+1*.1+2*.06+3*.03+4*.01=.36 E(A) = 1*.5+0*.5=.5 cov(M,A) = (0-.36)(0-.5)*.35 + (1-.36)(1-.5)*.065+ = -.105cov(M,A) = (0-....
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This note was uploaded on 11/23/2009 for the course FIN 5290 taught by Professor Li during the Fall '09 term at Temple.

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Lecture 2 - Probability distribution Outcomes: mutually...

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