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Lecture 2 - Probability distribution Outcomes mutually...

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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
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Normal distribution: The probability distribution function of a normal random variable X is: µ= location parameter , σ²= scale parameter (lower sigma indicates more ‘spread out’) We usually denote a normal random variable as X N(μ,σ²). When μ=0 and σ²=1, we have a standard normal distribution. We usually denote a standard normal random variable as z, with the cumulative distribution function being: P(Z≤z)=Φ(z) If X N(μ,σ²), then Y=(X-μ)/σ 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 ) = μ Y = long-run average value of Y over repeated realizations of Y variance = E ( Y μ Y )P 2 = 2 Y σ = measure of the squared spread of the Distribution standard deviation = variance = σ Y Example:
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Example: Example: If Y N(μ,σ²), then the mean of Y is μ, the variance of Y is σ². 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+… = -.105
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