Exam
Preliminary Examination 1998: Probability \& Statistics
1998 · 10 questions in this exam.
Questions in This Exam
Ordered by the question sequence in the original exam.
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Question 1
Suppose \(\{E_1,E_2,\ldots\}\) is a sequence of measurable (otherwise arbitrary) events. Suppose \(\{F_1,F_2,\ldots\}\) is another sequence of measurable events also totally independent of all of the \(E_i\)'s. Assume the following: **(a)** \(P(E_k\text{, infinitely often})=1\); **(b)** there exists some \(p>0\), such that for all \(k\geq 1\), \(P(F_k)\geq p\). Prove: \[ P(E_k\cap F_k\text{, infinitely often})\geq p>0. \] (10 points)
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Question 2
Let \(X_1,X_2,\ldots\) be independent identically distributed positive random variables with \(EX_1=\mu\) and \(\operatorname{var}(X_1)=\sigma^2\). Let \[ N(t)=\min\left\{k:\sum_{1\leq i\leq k}X_i>t\right\},\qquad 0
- Question 3
Let \(X_1,X_2,\ldots\) be independent random variables with distribution functions \(F_1,F_2,F_3,\ldots\). Let \(Y=\sup_{1\leq i<\infty}X_i\). **(a)** Show that \(P\{Y<\infty\}\) is 0 or 1 depending on whether \(\sum_{1\leq i<\infty}(1-F_i(x))\) converges for some \(x\). (5 points) **(b)** Show that if \(P\{Y<\infty\}=1\), then \(\prod_{1\leq i<\infty}F_i(x)\) converges for all \(x\) and it is the distribution function of \(Y\).- Question 4
Let \(X_1,X_2,X_3\) be i.i.d., each with an exponential distribution with mean 1. Find the joint distribution of \[ Y_1=\frac {X_1}{X_1+X_2},\qquad Y_2=\frac {X_1+X_2}{X_1+X_2+X_3},\qquad Y_3=X_1+X_2+X_3. \] (7 points) In particular, are they independent? (3 points)
- Question 5
Suppose \(N\) is a positive random variable. **(a)** Show that \[ P(N>0)\geq \frac{(EN)^2}{E N^2}. \] (5 points) **(b)** Let \(X_1,X_2,\ldots\) be i.i.d. random variables. Define \(S_n=X_1+\cdots+X_n\). Use part (a) to show that for any positive integer \(n\), \[ P(S_j=0\text{, for some }1\leq j\leq n) \geq \frac{\sum_{j=1}^{n}P(S_j=0)} {2\left(1+\sum_{j=1}^{n}P(S_j=0)\right)}. \] (5 points) ** Statistics**- Question 6
Let \(X_1,X_2,\ldots,X_n\) be independent identically distributed random variables with distribution function \(F\) and density function \(f\). We assume that the second derivative of \(f\) is bounded. We estimate \(f\) by \[ f_n(t)=\frac{F_n(t+h_n)-F_n(t-h_n)}{2h_n}, \] where \(F_n\) is the empirical distribution function and \(h_n\to0\) as \(n\to\infty\). **(a)** Show that \(Ef_n(t)\to f(t)\) as \(n\to\infty\). (5 points) **(b)** Show that if \(nh_n\to\infty\), then \[ \frac{f_n(t)-Ef_n(t)}{(\operatorname{var}(f_n(t)))^{1/2}} \] is asymptotically standard normal as \(n\to\infty\). (5 points)- Question 7
Let \(X_1,X_2,\ldots,X_n\) denote a random sample from a Poisson distribution with parameter \(\theta>0\). **(a)** Show that \((-1)^{X_1}\) is unbiased for \(e^{-2\theta}\), and use the Lehmann-Scheffe theorem to deduce a UMVUE of \(e^{-2\theta}\) based on the sample. (Hint: The conditional distribution of \(X_1\) given \(X_1+\cdots+X_n\) has a simple form.) (5 points) **(b)** Argue that the UMVUE in part (a) is a consistent estimator of \(e^{-2\theta}\). (5 points)- Question 8
Let \(X_1,\ldots,X_n\) be a random sample from the \(N(\mu,\sigma^2)\) distribution, with both \(\mu\) and \(\sigma^2\) unknown. **(a)** Derive the likelihood ratio test for \(H:\sigma^2=\sigma_0^2\) against all alternatives. Here \(\sigma_0^2\) is a known positive constant. (5 points) **(b)** Do the same for \(H:\sigma^2\geq\sigma_0^2\). (5 points)- Question 9
Let \(X_1,\ldots,X_n\) be a random sample from \(N(\theta,\theta)\) (i.e., normal with mean and variance both equal to \(\theta\)), where \(\theta>0\). **(a)** Give three pivotal quantities (or pivots) involving the entire sample, which respectively have a standard normal distribution, a \(t\)-distribution, and a chi-squared distribution, and indicate the numbers of degrees of freedom. (5 points) **(b)** Use the normal pivotal quantity in part (a) to obtain a \(100(1-\alpha)\) percent confidence interval for \(\theta\). (5 points)- Question 10
Let \(X_1,X_2,\ldots,X_n\) be independent Poisson random variables with parameters \(\theta_1,\theta_2,\ldots,\theta_n\). We wish to test \(H_0:\theta_1=\theta_2=\cdots=\theta_n\) against the alternative that \(H_0\) is not true. **(a)** Find the likelihood ratio test. (5 points) **(b)** Show that the likelihood ratio is asymptotically normal (after centralizing and normalizing) under \(H_0\). (5 points)Topics Covered by This Exam
Borel-Cantelli Lemmas Central Limit Theorems Conditional Expectation and Conditional Independence Convergence in Probability and Almost Sure Convergence Covariance and Independence Estimator Distributions F-Tests and t-Tests Laws of Large Numbers Likelihood Ratio Tests Maximum Likelihood Estimation Measure-Theoretic Probability Multivariate Normal Distributions Probability Inequalities and Tail Bounds Quadratic Forms and Chi-Squared Distributions Simultaneous Confidence Intervals Sufficiency, Completeness, and UMVUE Uniform Integrability and Truncation Methods - Question 3