Exam
Preliminary Examination \ 0.7 ] Probability \& Statistics \ 1.1 ] 1994
1994 · 11 questions in this exam.
Questions in This Exam
Ordered by the question sequence in the original exam.
-
Question 1
Let \(X_1,X_2,\cdots \) be independent, identically distributed random variables, uniformly distributed on \([0,1]\). Show that \[ Y_n=\frac {4\sum _{1\le k\le n}kX_k-n^2}{n^{3/2}}, \] converges in distribution to a normal random variable.
-
Question 2
Let \((\Omega ,\mathcal {A},P)\) be a probability space and \(\{B(t),0\leq t\leq 1\}\) be a Brownian motion on it. Since almost all sample paths of \(B\) are continuous, \(\int _0^t B(t)\,dt\) can be defined as a usual Riemann integral. Compute the distribution of \(\int _0^1 B(t)\,dt\).
-
Question 3
Let \(\{S_n,\mathcal {F}_n,n\geq 1\}\) be a nonnegative martingale with \(ES_n=1\). Show that for all \(\lambda >0\), \[ P\{S_n\geq \lambda ,\text { for some }n\geq 1\}\leq \frac {1}{\lambda }. \]
-
Question 4
Let \(X_1,X_2,\ldots\) be independent, identically distributed random variables. Show that the following statements are equivalent: **(a)** \(E|X_1|^\nu<\infty\); **(b)** \(X_n/n^{1/\nu}\to0\), almost surely; **(c)** \(\max_{1\leq i\leq n}|X_i|/n^{1/\nu}\to0\), almost surely.
-
Question 5
Let \(X_1,X_2,\cdots \) be independent, identically distributed normal random variables. Find two numerical sequences, \(a_n\) and \(b_n\), such that \[ \frac {\max _{1\le i\le n}X_i-a_n}{b_n} \] converges in distribution to a non-degenerate random variable.
-
Question 6
Let \(0\leq X_n\leq 1\) be adapted to \(\mathcal{F}_n\). Let \(\alpha,\beta>0\) with \(\alpha+\beta=1\) and suppose \[ P(X_{n+1}=\alpha+\beta X_n\mid \mathcal{F}_n)=X_n \] \[ P(X_{n+1}=\beta X_n\mid \mathcal{F}_n)=1-X_n. \] **(a)** Show that \(P\{\lim_{n\to\infty}X_n=0\text{ or }1\}=1\). **(b)** Show that if \(X_0=\theta\), then \(P\{\lim_{n\to\infty}X_n=1\}=\theta\).
-
Question 1-7
Suppose \(X_1,\ldots,X_m\) are independent with \(X_j\sim \operatorname{BIN}(n_j,p_j)\). **(a)** Find the UMVUE's of \(p_1,\ldots,p_m\). **(b)** Suppose you know that \(p_1=p_2=\cdots=p_m\). Let \(p\) denote this common (but unknown) value. Find the UMVUE of \(p\). **(c)** Find the likelihood ratio statistic, \(\lambda\), for \(H_0:p_1=\cdots=p_m\) versus \(H_1:p_i\neq p_j\), for some \(i\) and \(j\). **(d)** It can be shown that \(2\log\lambda\) is approximately the same as the usual \(\chi^2\) statistics. Using this fact, find an approximate test for \(H_0\) vs \(H_1\) above.
-
Question 2-8
Consider the linear model: \[ Y_{ij}=\beta_i+\epsilon_{ij},\qquad 1\leq i\leq 2,\qquad 1\leq j\leq J. \] Suppose \(\epsilon_{ij}\)'s are independent and for some (known) \(a_1\) and \(a_2\), \(\epsilon_{ij}\sim N(0,a_i^2\sigma^2)\). **(a)** Find the U.M.V.U.E.'s of \(\beta_1\) and \(\beta_2\). **(b)** Suppose you know that for some unknown \(\beta\), \(\beta_1=\beta_2\). Find the U.M.V.U.E. of \(\beta\).
-
Question 3-9
Suppose \(X\sim \operatorname {POISS}(\theta )\). Put a \(\operatorname {GAMMA}(\alpha ,\beta )\) prior on \(\theta \) and suppose we have the following loss function: \[ \ell (\theta ,a)=\frac {(a-\theta )^2}{\theta }. \] Find the Bayes’ estimator of \(\theta \).
-
Question 4-10
Suppose \(\theta \in \{\theta _0,\theta _1\}\) is an unknown (\(\theta _0\) and \(\theta _1\) are, however, known.) Put some prior, \(\pi \), on \(\theta \). We are to test \(H_0:\theta =\theta _0\) versus \(H_1:\theta =\theta _1\). Our actions are 0 (accept) and 1 (reject). Find the Bayes’ procedure for doing this test, if the loss function, \(\ell (\theta ,a)\), is the 0–1 loss given by: \[ \ell (0,1)=0,\qquad \ell (0,0)=1, \] \[ \ell (1,0)=0,\qquad \ell (1,1)=1. \] Is this procedure minimax?
-
Question 5-11
Construct a \((1-\alpha )\) two-sided confidence interval for the correlation coefficient of a bivariate normal distribution. (HINT. This is an exponential family.)