Exam
Preliminary Examination 1996: Probability \& Statistics
1996 · 10 questions in this exam.
Questions in This Exam
Ordered by the question sequence in the original exam.
-
Question 1
Prove Cantelli’s theorem: If \(X_1,X_2,\ldots \) are independent (but not necessarily identically distributed), mean zero, random variables with \(\sup _n E[X_n^4]<\infty \), then \(S_n/n\to 0\) a.s. as \(n\to \infty \).
-
Question 2
**(a)** Let \(X_1,X_2,\ldots\) be i.i.d. Poisson random variables with mean 1, and let \(Z\) be \(N(0,1)\). Denote \(a^-=-\min\{a,0\}\), and prove that \[ E\left[\left(\frac{S_n-n}{\sqrt{n}}\right)^-\right]\to E[Z^-]. \tag{*} \] **(b)** Evaluate both expectations in (*) explicitly, and, noting the telescoping sum, deduce Stirling's formula for \(n!\).
-
Question 3
Let \((\Omega ,\mathcal {F},P)\) be a probability space, let \(\mathcal {G}\) be a sub-\(\sigma \)-field of \(\mathcal {F}\), and let \(M\) be the closed subspace of \(L^2(\Omega ,\mathcal {F},P)\) consisting of the \(\mathcal {G}\)-measurable functions in \(L^2\). Show that \(T(X)=E[X\mid \mathcal {G}]\) coincides with the orthogonal projection of \(L^2(\Omega ,\mathcal {F},P)\) onto \(M\).
-
Question 4
Consider the Markov chain in the state space \(S=\{\ldots ,-2,-1,0,1,2,\ldots \}\) with transitions \(P(i,i+2)=p\) and \(P(i,i-1)=1-p\), where \(0<p<1\). Determine for which \(p\) this chain is recurrent and for which \(p\) it is transient.
-
Question 5
Let \(\{B(t),t\geq 0\}\) be a standard Brownian motion. Prove directly, using \[ V_n=\sum _{i=1}^{2^n}\left |B(i/2^n)-B((i-1)/2^n)\right |; \] that \(B(\cdot ,\omega )\) is of unbounded variation on \([0,1]\) for a.e. \(\omega \).
Statistics
-
Question 6
Suppose \(X_1,\ldots,X_n\) is an i.i.d. sample from a normal population with \[ EX_1=\operatorname{Var}(X_1)=\mu>0. \] **(a)** Compute the maximum likelihood estimator \(\hat{\mu}\) of \(\mu\); **(b)** Is \(\hat{\mu}\) consistent? **(c)** Is \(\hat{\mu}\) asymptotically normal?
-
Question 7
Suppose \(X_1,\ldots,X_n\) is an i.i.d. sample from a normal distribution with mean \(\mu\) and variance 1. **(a)** Find the UMVU estimator for \(\mu\). (Prove the optimality criterion.) **(b)** Put a \(N(\theta,\tau^2)\) prior on \(\mu\) and find the minimax estimator of \(\mu\).
-
Question 8
Consider the linear model: \(Y_i=\beta+\epsilon_i\), \(1\leq i\leq n\). Here, \(\epsilon_i\)'s are i.i.d. \(N(0,\sigma^2)\), and \(\sigma\) and \(\beta\) are unknown. **(a)** Find the least squares estimator of \(\beta\); **(b)** Find the UMVU estimator for \(\beta\) (prove the optimality); **(c)** Find the UMP test for \(H_0:\beta=\beta_0\) versus \(H_1:\beta=\beta_1\); **(d)** Discuss--without proofs--how to find the UMP test for \(H_0:\beta>\beta_0\) vs. \(H_1:\beta\leq\beta_0\) from tests of the form in part (c) above.
-
Question 9
Let \(X_1,\ldots,X_n\) be an i.i.d. sample which is uniformly chosen from the interval \((\theta_1,\theta_2)\). **(a)** Prove that there are no one-dimensional (i.e., not vector-valued) sufficient statistics for \(h(\theta_1,\theta_2)\) where \(h\) is a one-to-one measurable function. **(b)** Is there a one-dimensional (i.e., not vector-valued) sufficient statistics for \(\mu=(\theta_1+\theta_2)/2\)?
-
Question 10
Let \(X_1,\ldots,X_n\) be an i.i.d. sample from a uniform \((0,\theta)\) distribution, where \(\theta>0\). **(a)** Find the maximum likelihood estimator \(\hat{\theta}\) for \(\theta\); **(b)** Construct a \(100(1-\alpha)\%\) confidence interval for \(\theta\) based on \(\hat{\theta}\).