Advanced Level Mathematics: Statistics 2 by Steve Dobbs, Jane Miller, Julian Gilbey

By Steve Dobbs, Jane Miller, Julian Gilbey

Written to check the contents of the Cambridge syllabus. records 2 corresponds to unit S2. It covers the Poisson distribution, linear combos of random variables, non-stop random variables, sampling and estimation, and speculation exams.

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1. Inverse Gamma Distribution ????????(????, ????) The pdf to the inverse gamma distribution is: −???? ???????? ????(????; ????, ????) = . ???? ???? . 2. Student T Distribution ????(????, ????, ???????? ) The pdf to the standard student t distribution with ???? = 0, ???? 2 = 1 is: − ????2 ????(????; ????) = . �1 + � ???? √????????Γ(????⁄2) Γ[(???? + 1)⁄2] ????+1 2 The generalized student t distribution is: − (???? − ????)2 ????(????; ????, ????, ???? 2 ) = . 3. F Distribution ????(???????? , ???????? ) ????+1 2 ????=???? Also known as the Variance Ratio or Fisher-Snedecor distribution the pdf is: With cdf: ????(????; ????) = ???????? � (????1 ????)????1 .

292). 292). The test statistic for the likelihood ratio is: ???? = 2[Λ�????�� − Λ(????)] ???? is approximately Chi-Square distributed with one degree of freedom. ???? = 2�Λ�????�� − Λ(????)� ≤ ???? 2 (????; 1) Where ???? is the 100????% confidence interval for ????. The two sided confidence ???????? are calculated by solving: limits ???????? and ��� Λ(????) = Λ�????�� − ???? 2 (????; 1) 2 The limits are normally solved numerically. The likelihood ratio intervals are always within the limits of the parameter and gives asymmetrical confidence limits.

7. Obtained the desired information. This may be the distribution parameters or estimates of reliability or hazard rate trends. When multiple failure modes are observed only one failure mode should be plotted with the other failures being treated as censored. Two popular methods to treat censored data two methods are : Rank Adjustment Method. (Manzini et al. 140) Here the adjusted rank, ???????????? is calculated only for non-censored units (with ???????????? still being the rank for all ordered times). This adjusted rank is used for step 2 with the remaining steps unchanged: (???? + 1) − ????????????−1 ???????????? = ????????????−1 + 2 + ???? − ???????????? Kaplan Meier Estimator.

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