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The final exam solutions. Part I, #1, Central limit theorem. Let X1,X2, … , Xn be a sequence of i.i.d. random variables each having mean μ and variance σ2 The sum of a large number of independent random variables has a distribution that is approximately normal. Part II #2.
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Part I, #1, Central limit theorem • Let X1,X2, …, Xn be a sequence of i.i.d. random variables each having mean μ and variance σ2 • The sum of a large number of independent random variables has a distribution that is approximately normal
Part II #2 • If the successful probability of trial is very small, then the accumulatively many successful trials distribute as a Poisson. • The Poisson parameter λ is the mean frequency of interest within a large bundle of experiment • The exponential distribution is often used to describe the distribution of the amount of time until the specific event first occurs. • The exponential distribution seems to partition the Poisson distribution with a parameter λ by every two occurrences into several small time intervals and focuses on a specific time interval. • The same distributional parameter λ, • The exponential distribution emphasizes the cycle time, 1/λ, while the Poisson on the frequency λ.
Part I, #3 If Z and Xn2 are independent random variables, with Z having a std. normal dist. And Xn2 having a chi-square dist. with n degrees of freedom, • For large n, the t distribution approximates to the standard normal distribution
Part I, #4 • When these two population variances are unknown but equal, we calculate the pooled variance estimator, Sp2, by means of weighted average of individual sample variance, S12 and S22
Part I, #5 • It is not correct. • He had better say that the statistic or random variables used to obtain this confidence interval is such that 95% of the time that it is employed it will result in an interval covered the μ. • He can assert with probability 0.95 the interval will cover the μ only before the sample data are observed. • Whereas after the sample data are observed and computed the interval, he can only assert that the resultant interval indeed contains μ with confidence 0.95.
Part I, #7 • (a) if d(X) is the estimator of θ, and E[d(X)]=θ, then the d(x) is unbiased. • (b) the MLE estimator is not unbiased. • (c) the sample variance is an unbiased estimator of σ2
Part II #1 • Mean=np=90, • Var=np(1-p)=150(.6)(.4)=36, ∴standard deviation=6 • P{X≦80}=p{X<80.5}=P{(X-90)/6<(80.5-90)/6}=P{Z<-1.583} =1-P{Z≦1.583} (=1-0.943=0.057)
