Preliminary Examination 1998: Probability \& Statistics

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) \(\mathbb {P}(E_k\text {, infinitely often})=1\);

(b) there exists some \(p>0\), such that for all \(k\ge 1\), \(\mathbb {P}(F_k)\ge p\).

Prove: \(\mathbb {P}(E_k\cap F_k\text {, infinitely often})\ge p>0\). (10 points)

Let \(X_1,X_2,\ldots \) be independent identically distributed positive random variables with \(\E X_1=\mu \) and \(\operatorname {Var}(X_1)=\sigma ^2\). Let \[ N(t)=\min \left \{k:\sum _{1\le i\le k}X_i>t\right \},\qquad 0<t<\infty . \]

(a) Show that \[ \lim _{t\to \infty }\frac {1}{t}N(t)=\frac {1}{\mu } \] almost surely. (5 points)

(b) Show that \((N(t)-t/\mu )/(t\sigma ^2/\mu ^3)^{1/2}\) goes in distribution to a standard normal random variable. (5 points)

Let \(X_1,X_2,\ldots \) be independent random variables with distribution functions \(F_1,F_2,F_3,\ldots \). Let \(Y=\sup _{1\le i<\infty }X_i\).

(a) Show that \(\mathbb {P}\{Y<\infty \}\) is 0 or 1 depending on whether \(\sum _{1\le i<\infty }(1-F_i(x))\) converges for some \(x\). (5 points)

(b) Show that if \(\mathbb {P}\{Y<\infty \}=1\), then \(\prod _{1\le i<\infty }F_i(x)\) converges for all \(x\) and it is the distribution function of \(Y\).

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)

Suppose \(N\) is a positive random variable.

(a) Show that \[ \mathbb {P}(N>0)\ge \frac {(\mathbb {E} N)^2}{\mathbb {E} N^2}. \] (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 \(n\), \[ \mathbb {P}(S_j=0,\text { for some }1\le j\le n) \ge \frac {\sum _{j=1}^n\mathbb {P}(S_j=0)} {2\left (1+\sum _{j=1}^n\mathbb {P}(S_j=0)\right )}. \] (5 points)

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 \(h_n\to 0\) as \(n\to \infty \).

(a) Show that \(\E f_n(t)\to f(t)\) as \(n\to \infty \). (5 points)

(b) Show that if \(nh_n\to \infty \), then \[ \frac {f_n(t)-\E f_n(t)}{(\operatorname {Var}(f_n(t)))^{1/2}} \] is asymptotically standard normal as \(n\to \infty \). (5 points)

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–Scheffé 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)

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 \(H:\sigma ^2\ge \sigma _0^2\). (5 points)

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)

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 \(H_o:\theta _1=\theta _2=\cdots =\theta _n\) against the alternative that \(H_o\) 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 \(H_o\). (5 points)