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Ch 4 & 5 Important Ideas

This text covers important concepts in probability, including sampling theory, density integration, normal distribution, gamma distribution, jointly distributed random variables, correlation, and covariance.

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Ch 4 & 5 Important Ideas

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  1. Ch 4 & 5 Important Ideas Sampling Theory

  2. Density Integration for Probability (Ch 4 Sec 1-2) • Integral over (a,b) of density is P(a<X<b) • P(X≤x) =FX(x) is the cdf and d/dx(FX(x)) = fX(x) the pdf (density) • E(X) = integral of (x fX(x)) over all x. • E(X2) = integral of (x2 fX(x)) over all x. • Examples of calculations posted. • Definition of Variance and SD (p 158)

  3. Normal Distribution Ch 4 Sec 3 • Common for measurements • Defined once mean and SD known • Probabilities of N(0,1) are tabulated • N(,) probabilities can be related to N(0,1) • 68%,95%,99% within 1,2,3 SDs of mean (approx) • Close link with lognormal

  4. Gamma Distribution Ch 4 Section 4 • Parameters ,  describes shape and  scale. Mean=  SD = 1/2 • Models waiting time until th event • Exponential and Chi-squared are special cases. • cdf is tabulated (like normal - no closed form, except for =1)

  5. Jointly Distributed RVs Ch 5 Sec 1 • Independence • Joint, Marginal, Conditional Distr’ns - Discrete and Continuous RVs • Exercises posted (5.1.1, 5.1.12, 5.1.15)

  6. Correlation and CovarianceCh 5 Section 2 • V(X+ Y) = V(X)+ V(Y) + 2Cov(X,Y) • Cov(X,Y) = E((X-x)(Y-y)) • Corr(X,Y) = Cov(X,Y)/(SD(X)SD(Y)) • -1 < Corr < +1

  7. Sampling TheoryCh 5 Sec 3,4 • Random Sampling (p 228) • Sampling Distribution of a Statistic • Sampling Distribution of the Sample Mean, • Square Root Law • Central Limit Theorem (Averages and Sums tend to Normality) has an approx N dist for large n

  8. Using the CLT • When pop mean and SD knownSee notes re risky company mean = 0.38, SD = 1.56 • When pop mean and SD estimated(don’t need to know mean and SD)(don’t need to know pop distr’n)

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