Probability Prelim 1.3 January 7, 2010

In this problem, all random variables are nonnegative. We say that \(X\) is stochastically dominated by \(Y\) if \(\mathbb {P}\{X>a\}\leq \mathbb {P}\{Y>a\}\) for all \(a>0\). Prove that if \(X\) is stochastically dominated by \(Y\), then \(\mathbb {E}\Phi (X)\leq \mathbb {E}\Phi (Y)\) for all increasing functions \(\Phi :\mathbf {R}_+\to \mathbf {R}_+\).

Suppose \((\Omega ,\mathcal {F},\mathbb {P})\) is a probability space, and \(\mathcal {F}_1\subset \mathcal {F}_2\subset \cdots \) defines a filtration of sigma-algebras of subsets of \(\mathcal {F}\).

(a) State, without proof, Doob’s martingale convergence theorem.

(b) Prove that \(\mathcal {L}:=\lim _{n\to \infty }\mathbb {E}(Z\mid \mathcal {F}_n)\) exists almost surely and in \(L^1(\mathbb {P})\) for all \(Z\in L^1(\mathbb {P})\) [this is Lévy’s martingale convergence theorem]. Identify the limit \(\mathcal {L}\).

Suppose \(X_1,X_2,\ldots \) are independent, identically-distributed exponential random variables with mean \(\lambda >0\). Prove that \[ \max (X_1\ldots ,X_n)-\frac {1}{\lambda }\ln n\Rightarrow X, \] and compute \(\mathbb {P}\{X>x\}\) for all \(x>0\).

Give a rigorous proof that \(\mathbb {E}[XY]=\mathbb {E}[X]\mathbb {E}[Y]\) if \(X\) and \(Y\) are independent random variables belonging to \(L^1(\mathbb {P})\). In particular, show that \(XY\in L^1(\mathbb {P})\).

Fix \(n\geq 2\) and let \(X,Y_1,\ldots ,Y_n\) be jointly distributed random variables. We say that \(Y_1,\ldots ,Y_n\) are conditionally i.i.d. given \(X\) if \[ \mathbb {P}(Y_1\leq y_1,\ldots ,Y_n\leq y_n\mid X) =\mathbb {P}(Y_1\leq y_1\mid X)\cdots \mathbb {P}(Y_1\leq y_n\mid X) \]

for all \(y_1,\ldots ,y_n\). Show that, if \(Y_1,\ldots ,Y_n\) are conditionally i.i.d. given \(X\), then \[ \operatorname {Var}(Y_1+\cdots +Y_n)=n^2\operatorname {Var}(Y_1)-n(n-1)\mathbb {E}[\operatorname {Var}(Y_1\mid X)]. \]

A random experiment has exactly three possible outcomes, referred to as outcomes 1, 2, and 3, with probabilities \(p_1>0\), \(p_2>0\), and \(p_3>0\), where \(p_1+p_2+p_3=1\). We consider a sequence of independent trials, at each of which the specified random experiment is performed. For \(i=1,2\), let \(N_i\) be the number of trials needed for outcome \(i\) to occur, and put \(N:=\min (N_1,N_2)\).

(a) Show that \(N\) is independent of \(1_{\{N_1<N_2\}}\).

(b) Evaluate \(\mathbb {E}[N_1\mid N_1<N_2]\).

(c) Roll a pair of dice until a total of 6 or 7 appears. Given that 6 appears before 7, what is the (conditional) expected number of rolls?

If \(X\) is either (a) Poisson\((\lambda )\) or (b) gamma\((\lambda ,1)\) (density proportional to \(x^{\lambda -1}e^{-x}\), \(x>0\)), show that \((X-\mathbb {E}[X])/\sqrt {\operatorname {Var}(X)}\) converges in distribution to \(N(0,1)\) as \(\lambda \to \infty \) (\(\lambda \) need not be an integer).

Let \(X_1,X_2,\ldots \) be i.i.d. with mean \(\mu \) and finite variance. Show that \[ U_n:=\binom {n}{2}^{-1}\sum _{1\leq i<j\leq n}X_iX_j \] converges in probability to \(\mu ^2\) as \(n\to \infty \).

Let \(X_1,X_2,\ldots \) be i.i.d. with mean \(\mu \). Define \(S_n:=X_1+\cdots +X_n\) for all \(n\geq 1\). For fixed \(n\geq 2\), define \[ M_1:=\frac {S_n}{n},\qquad M_2:=\frac {S_{n-1}}{n-1},\qquad \ldots \qquad M_n:=\frac {X_1}{1}. \] (a) Show that \(\mathbb {E}[X_k\mid S_n]=S_n/n\) for \(1\leq k\leq n\).

(b) Show that \(M_1,M_2,\ldots ,M_n\) is a martingale.

Let \(Z\) be a random variable with all moments finite. Choose \(X\) and \(Y\) appropriately as in the Cauchy–Schwarz inequality or the Hölder inequality to prove that \(f(p):=\ln \mathbb {E}[|Z|^p]\) is convex on \((0,\infty )\).