Probability Qualifying Examination 1.3 January 8, 2009

Let \(X_1,X_2,\ldots \) be i.i.d. with mean \(\mu \) and variance \(\sigma ^2\), and put \(S_0:=0\) and \(S_n:=X_1+\cdots +X_n\) for each \(n\geq 1\). Let \(N\) be a nonnegative-integer-valued random variable independent of \(X_1,X_2,\ldots \). Find the mean and variance of \(S_N\) by conditioning on \(N\).

(a) Find a closed-form expression (i.e., one with no summation sign) for \(\mathbb {E}[(X+1)^{-1}]\) when \(X\) is binomial\((n,p)\) with \(n\geq 1\) and \(0<p<1\).

(b) Find a closed-form expression for \(\mathbb {E}[(X+1)^{-1}]\) when \(X\) is Poisson\((\lambda )\) with \(\lambda >0\). You may use the result of part (a) here.

For \(n=1,2,\ldots \), let \(X_n\) have density \[ f_n(x):=\frac {n}{\pi (1+n^2x^2)},\qquad -\infty <x<\infty . \] (a) Does \(X_n\) converge to 0 in probability?

(b) Does \(X_n\) converge to 0 in \(L^1(\mathbb {P})\)?

(c) If \(X_1,X_2,\ldots \) are independent, does \(X_n\) converge to 0 a.s.?

Let \(X_1,X_2,X_3,Y_1,Y_2,Y_3\) be independent \(N(0,1)\) random variables. Let \(P_1\), \(P_2\), and \(P_3\) be the points in the plane with coordinates \((X_1,Y_1)\), \((X_2,Y_2)\), and \((X_3,Y_3)\). Let \(M\) be the midpoint of the segment \(P_1P_2\) and let \(r\) be half its length. Show that the probability that the point \(P_3\) lies within the circle of radius \(r\) centered at \(M\) is equal to \(1/4\).

Let \(X_1,X_2,\ldots \) be an i.i.d. sequence of integer-valued random variables, and let \(\lambda _0\) be a root of the equation \(\mathbb {E}[\lambda ^{X_1}]=1\). (If \(\lambda _0\) is complex, it suffices to notice that expectations, conditional expectations, and martingales extend easily to complex-valued random variables.) Let \(S_n:=X_1+\cdots +X_n\) and \(M_n:=\lambda _0^{S_n}\) for each \(n\geq 0\), where \(S_0:=0\).

(a) Show that \(\{M_n\}_{n\geq 0}\) is a martingale with respect to \(\mathcal {F}_n:=\sigma (X_1,\ldots ,X_n)\).

(b) Consider the special case that \(\mathbb {P}\{X_1=1\}=p\) and \(\mathbb {P}\{X_1=-1\}=1-p\), where \(0<p<1\). Show that, in this case, \(\lim _{n\to \infty }M_n\) exists a.s. and is finite.

(c) Find another example to show that \(\lim _{n\to \infty }M_n\) need not exist a.s.

Suppose \(X_n\Rightarrow X\) as \(n\to \infty \), and \(\sup _{n\geq 1}\mathbb {E}(X_n^2)<\infty \). Prove that \(X\in L^1(\mathbb {P})\) and \(\lim _{n\to \infty }\mathbb {E} X_n=\mathbb {E} X\).

Choose and fix a sequence of real numbers \(a_1,a_2,\ldots \) such that: (i) \(0<a_k<1\) for all \(k\geq 1\); and (ii) \(\sum _{k=1}^{\infty }a_k^2=\infty \). Let \(X_1,X_2,\ldots \) be independent random variables, where each \(X_k\) is distributed uniformly on \((-a_k,a_k)\). Prove that \(n^{-1/2}(X_1+\cdots +X_n)\) converges weakly to \(N(0,\sigma ^2)\) and compute \(\sigma \).

Suppose \(f\) is a probability density function on \(\mathbf {R}\), and define for all \(x,y\in \mathbf {R}\), \[ g(x,y):= \begin {cases} 2f(x)f(y), & \text {if } x\leq y,\\ 0, & \text {otherwise.} \end {cases} \] (a) Prove that \(g\) is the probability density function of some random variable \((X,Y)\) which takes values in \(\mathbf {R}^2\).

(b) Are \(X\) and \(Y\) independent? Prove or disprove.

Suppose \(X_1,X_2,\ldots \) are independent, identically-distributed random variables such that \[ \mathbb {P}\{X_1>x\}= \begin {cases} x^{-\alpha }, & \text {if } x\geq 1,\\ 1, & \text {if } x<1, \end {cases} \] where \(\alpha >0\) is fixed. Prove that \[ n^{-1/\alpha }\max (X_1,\ldots ,X_n)\Rightarrow X, \] and compute \(\mathbb {P}\{X>x\}\) for all \(x\in \mathbf {R}\).

Let \(\alpha ,\lambda \) be fixed, strictly positive, and finite. The Gamma\((\alpha ,\lambda )\) density function is \[ f(x):= \begin {cases} \dfrac {\lambda ^\alpha }{\Gamma (\alpha )}x^{\alpha -1}e^{-\lambda x}, & \text {if } x>0,\\ 0, & \text {if } x\leq 0, \end {cases} \] where \(\Gamma (\alpha ):=\int _0^\infty t^{\alpha -1}e^{-t}\,dt\). This defines a probability density function.

(a) Compute the characteristic function of \(f\).

(b) Suppose \(X_1,\ldots ,X_n\) are independent with common density Gamma\((1,\lambda )\). Prove that \(S_n:=X_1+\cdots +X_n\) has a Gamma density; also compute its parameters.

(c) Compute \(\mathbb {P}\{S_n<k<S_{n+1}\}\) for all integers \(k\geq 1\).