Probability Qualifying Examination 1.3 August 18, 2009

Let \(X\) be an exponential random variable with mean 1, that is, \(\mathbb {P}(X>u)=e^{-u}\) for all \(u>0\). Evaluate \(\mathbb {E}[X\mid X\wedge t]\) and \(\mathbb {E}[X\mid X\vee t]\) for all \(t>0\). Here \(a\wedge b:=\min (a,b)\) and \(a\vee b:=\max (a,b)\).

Let \(X\) be a random variable with values in an interval \(I\), and suppose that \(f\) and \(g\) are nondecreasing functions on \(I\) such that \(f(X)\) and \(g(X)\) have finite variance. Show that \(\operatorname {Cov}(f(X),g(X))\geq 0\).

Hint: If \(X_1\) and \(X_2\) are i.i.d. as \(X\), then \((f(X_1)-f(X_2))(g(X_1)-g(X_2))\geq 0\).

Let \(X_1,X_2,\ldots \) be independent with \[ \mathbb {P}(X_n=n^2-1)=n^{-2}=1-\mathbb {P}(X_n=-1),\qquad n\geq 1. \] Notice that \(\mathbb {E}[X_n]=0\) for all \(n\geq 1\), which might lead one to expect that \(S_n:=X_1+\cdots +X_n\) satisfies \(S_n/n\to 0\) a.s. Show that in fact \(S_n/n\to -1\) a.s.

Let \(X_1,X_2,\ldots \) be an i.i.d. sequence with \(\mathbb {P}(X_1=1)=p\) and \(\mathbb {P}(X_1=0)=1-p\), where \(0<p<1\). For each \(m\geq 1\), define \[ N_m:=\min \{n\geq m:(X_{n-m+1},\ldots ,X_n)=(1,1,\ldots ,1)\} \] and show that \[ \mathbb {E}[N_m]=\frac {1}{p}+\frac {1}{p^2}+\cdots +\frac {1}{p^m}. \] Hint: Evaluate \(\mathbb {E}[N_m]\) by conditioning on \(N_{m-1}\).

Consider a deck of \(n\) cards labeled \(1,2,\ldots ,n\). Assume it is well shuffled with every possible arrangement equally likely. Let \(E_n\) be the event that card \(j\) is in position \(j\) in the shuffled deck for some \(j\in \{1,2,\ldots ,n\}\). Evaluate \(\mathbb {P}(E_n)\) using inclusion-exclusion and show that \(\lim _{n\to \infty }\mathbb {P}(E_n)=1-e^{-1}\).

The Cauchy distribution has density \(f(x):=1/[\pi (1+x^2)]\) for \(-\infty <x<\infty \) and characteristic function \(\varphi (t):=e^{-|t|}\) for \(-\infty <t<\infty \). Let \(X_1,X_2,\ldots \) be i.i.d. Cauchy, put \(S_n:=X_1+\cdots +X_n\) for each \(n\geq 1\). Show that \(S_n/n\) does not converge almost surely or in probability, but does converge in distribution.

Let \(X_1,X_2,\ldots \) be i.i.d. uniform\((0,2)\), and put \(M_n:=X_1X_2\cdots X_n\) for each \(n\geq 1\). Show that \(\{M_n\}_{n=1}^{\infty }\) is a martingale. What does the martingale convergence theorem tell us about \(\lim _{n\to \infty }M_n\)? (In particular, evaluate this limit.)

Suppose \(X_1\) and \(X_2\) are independent random variables that satisfy the following three properties: (i) \(Y_1:=(X_1+X_2)/\sqrt {2}\) is standard normal; (ii) \(Y_2:=(X_1-X_2)/\sqrt {2}\) is standard normal; and (iii) \(Y_1\) and \(Y_2\) are independent. Prove that \(X_1\) and \(X_2\) are normally distributed. Can you compute their respective means and variances?

Prove that if \(\mathbb {P}\{X\geq 0\}=1\) and \(0<\mathbb {E}(X^2)<\infty \), then \[ \mathbb {P}\{X=0\}\leq \frac {\operatorname {Var} X}{\mathbb {E}(X^2)}. \]

Let \(X\) and \(Y\) be independent, standard-normal random variables. Find the distribution of \(X/Y\)?

Let \(X_1,X_2,\ldots \) be a sequence of i.i.d. random variables, each taking the values 0 and 1 with equal probabilities \(1/2\). Define \[ U:=\sum _{n=1}^{\infty }\frac {X_n}{2^n}. \] Prove that \(U\) is distributed uniformly on \((0,a)\) and compute \(a\).