Ch 4 & 5 Important Ideas Sampling Theory. Density Integration for Probability (Ch 4 Sec 1-2) Integral over (a,b) of density is P(a<X<b) P(X≤x) =F X (x)

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Presentation transcript:

Ch 4 & 5 Important Ideas Sampling Theory

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

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

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)

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

Correlation and Covariance Ch 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

Sampling Theory Ch 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

Using the CLT When pop mean and SD known See 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)