課程名稱︰統計導論
課程性質︰數學系選修
課程教師︰江金倉
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2016.12.7
考試時限（分鐘）：50
試題 :
Introduction to Statistics (Test3)
1.(10%) Are two independent events mutually exclusive? Explain your answer.
2. In the AIDS study, let D and T denote the events of HIV+ patients and
patients who are diagnosed as HIV+ patients, respectively. Suppose that the
sensitivity P(T|D) and the specificity P(T'|D') are known.
(2a)(10%) Compute the odds ratio of D versus D'.
(2b)(5%) What additonal condition is required to compute the positive
predictivity P(D|T) and the negative predictivity P(D'|T')?
3.(10%) Suppose that P(D)～0 in a case-control study. Show that the relative
risk of E versus of E' can approximated by the odds ratio of E versus E'.
4.(7%)(8%) An experiment consists of a sequence of independent coin tosses.
Let X denote the number of heads occurring within n tosses and Y be the number
of tails occurring before the rth head. Write the probability density function
of X and Y.
5. Let T be the waiting time until an event occurs of the Poisson process
{N_t:t>0} with P(N_t=m)={(λt)^m exp(-λt)/(m!)}1_{0,1,...,n,...}(m).
(5a)(8%) Derive the density function of T.
(5b)(7%) Show that T has a memoryless property.
6.(10%) Let X_1,...,X_n be a random sample from a Bernoulli(p), 0<p<1. Assume
that the sample size is large enough. Find an approximated probability of
P(Σ_{i=1}^{n} X_i<x) with continuity correction, wherex=1,2,...,n.
7.(5%)(10%) Let X_1,X_2,...,X_n be a random sample from a finite population
{x^(1),...,x^(N)}. Find an unbiased estimator of mean μ=Σ_{i=1}^{N} x^(i)/N
and variance σ^2=Σ_{i=1}^{N} (x^(i)-μ)^2/N.
8.(4%)(6%) Let X and Y be random variables with E[X]=μ_x, E[Y]=μ_y, Var(X)=
σ_x^2, Var(Y)=σ_y^2, and Cov(X,Y)=σ_xy^2. What are the mean and variance of
aX-bY for any constants a and b?