Math Problems Vault

Problem 10.1

If the events $A_{1}, \ldots , A_{n}$ are independent, then so are the events $A_{1}^{\prime }, \ldots , A_{n}^{\prime }$ where $A_{i}^{\prime }$ is either $A_{i}$ or $A_{i}^{c}, i = 1, \ldots , n$ .

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Problem 10.2

Consider the measurable space $(\Omega , \mathcal{A})$ , let $\mathcal{F}_{i}, i = 1,2$ , be fields with $\mathcal{F}_{i} \subseteq \mathcal{A}, i = 1,2$ , and define $\mathcal{F}$ by

\begin{aligned} \mathcal{F} = \left\{ \text{ all finite unions of } A \mathrm{~ s} \text{ with } A \in \mathcal{A} ; A = A_{1} \cup A_{2} \text{ or } A = A_{1} \cap A_{2}, A_{i} \in \mathcal{F}_{i}, i = 1,2\right\} . \end{aligned}

Then show that $\mathcal{F}$ is a field.

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Problem 10.3

For any real numbers $p_{1}, \ldots , p_{n}$ such that $0 \leq p_{i} \leq 1, i = 1, \ldots , n$ , show that

1-\exp \left(-\sum _{i = m}^{n} p_{i}\right) \leq 1-\prod _{i = m}^{n}\left(1-p_{i}\right) \leq \sum _{i = m}^{n} p_{i}, \quad m = 1, \ldots , n .

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Problem 10.4

(i)(ii)

Let $A_{n}, n = 1,2, \ldots$ , be independent events in the probability space $(\Omega , \mathcal{A}, P)$ . Then show that

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(i)

$\sum_{n = 1}^{\infty } \mathbb {P}\left(A_{n}\right) = \infty$ if and only if $\mathbb {P}\left(\limsup_{n \rightarrow \infty } A_{n}\right) = 1$ .

(ii)

$\sum_{n = 1}^{\infty } \mathbb {P}\left(A_{n}\right) < \infty$ if and only if $\mathbb {P}\left(\lim \sup_{n \rightarrow \infty } A_{n}\right) = 0$ .

Remark.

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This result is referred to as the Borel Zero-One Criterion.

Problem 10.5

Let $A_{n}, n = 1,2, \ldots$ , be independent events in the probability space $(\Omega , \mathcal{A}, P)$ , and suppose that $\lim_{n \rightarrow \infty } A_{n}$ exists, call it $A$ . Then show that $\mathbb {P}\left(A\right) = 0$ or $\mathbb {P}\left(A\right) = 1$ .

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Problem 10.6

Let $X_{n}, n \geq 1$ , be independent r.v.s distributed as $B(1, p)$ , and set $\bar{X}_{n} = n^{-1} \sum_{i = 1}^{n} X_{i}$ . Then show that $\bar{X}_{n} \xrightarrow [n \rightarrow \infty ]{P} p$ and $\bar{X}_{k^{2}} \underset {k \rightarrow \infty }{\text{ a.s. }} p$ .

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Problem 10.7

(i)(ii)

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(i)

If $X_{n}, n = 1,2, \ldots$ , are independent r.v.s defined on the probability space $(\Omega , \mathcal{A}, P)$ , show that $X_{n} \xrightarrow [n \rightarrow \infty ]{\text{ a.s. }} 0$ if and only if $\sum_{n = 1}^{\infty } \mathbb {P}\left(\left|X_{n}\right| \geq \varepsilon \right) < \infty$ for every $\varepsilon > 0$ .

(ii)

Reconcile this result with the result in Exercise 4 in Chapter 3.

Problem 10.8

Refer to Exercise 11 in Chapter 9 and construct a concrete example, by means of independent r.v.s (e.g., Binomially distributed r.v.s), to demonstrate the correctness of the assertion made there.

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Problem 10.9

The r.v.s $X_{1}, \ldots , X_{k}$ are independent if and only if in relation (10.4) all $B_{j} \mathrm{~ s}$ are replaced by intervals $\left(a_{j}, b_{j}\right]$ with $a_{j}, b_{j}$ in $\mathbb {R}$ and $a_{j} < b_{j}, j = 1, \ldots , k$ .

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Problem 10.10

Let $X$ and $Y$ be independent r.v.s and suppose that $\mathbb {E}\left[X\right]$ exists. For every $B \in \mathcal{B}$ , let $A = Y^{-1}(B)$ and show that $\int_{A} X d P = (\mathbb {E}\left[X\right]) \mathbb {P}\left(A\right)$ .

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Problem 10.11

Show that a r.v. $X$ is independent of itself if and only if $\mathbb {P}\left(X = c\right) = 1$ for some (finite) constant $c$ .

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Problem 10.12

(i)(ii)

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(i)

For two r.v.s with finite second moments, it follows that, if $X$ and $Y$ are independent, then they are uncorrelated (i.e., $\rho (X, Y) = 0$ , or equivalently, $\operatorname {Cov}\left[ X, Y \right] = 0)$ . Justify this statement.

(ii)

For the case that the r.v.s $X$ and $Y$ have the Bivariate Normal distribution, use Exercise 14(iii) in Chapter 9 in order to show that, if $X$ and $Y$ are uncorrelated, then they are independent.

Problem 10.13

If the r.v.s $X_{1}$ and $X_{2}$ are independent, then show that

\mathbb {E}\left[e^{i t\left(X_{1}+X_{2}\right)}\right] = \mathbb {E}\left[e^{i t X_{1}}\right] \times \mathbb {E}\left[e^{i t X_{2}}\right], \quad t \in \mathbb {R},

and by induction,

\mathbb {E}\left[e^{i t\left(X_{1}+\cdots +X_{k}\right)}\right] = \mathbb {E}\left[e^{i t X_{1}}\right] \times \cdots \times \mathbb {E}\left[e^{i t X_{k}}\right], \quad t \in \mathbb {R},

for the independent r.v.s $X_{1}, \ldots , X_{k}$ .

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Problem 10.14

(i)(ii)

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(i)

If the r.v.s $X$ and $Y$ are independent, distributed as $N\left(0, \sigma^{2}\right)$ , and $U = X+Y, V = X-Y$ , then show that $U$ and $V$ are independent, distributed as $N\left(0,2 \sigma^{2}\right)$ , by transforming the joint p.d.f. of $X$ and $Y$ into the joint p.d.f. of $U$ and $V$ .

(ii)

If the r.v.s $X$ and $Y$ are independent, distributed as $N\left(\mu_{1}, \sigma^{2}\right)$ and $N\left(\mu_{2}, \sigma^{2}\right)$ , respectively, use part (i) in order to show that $U$ and $V$ are independent, distributed as $N\left(\mu_{1}+\mu_{2}, 2 \sigma^{2}\right)$ and $N\left(\mu_{1}-\mu_{2}, 2 \sigma^{2}\right)$ , respectively.

Problem 10.15

(i)(ii)

Consider the probability space $(\Omega , \mathcal{A}, P)$ , and let $A_{1}, \ldots , A_{n}$ be independent events with $\mathbb {P}\left(A_{k}\right) = p, k = 1, \ldots , n$ . Next, define the function $X: \Omega \rightarrow \mathbb {R}$ as follows:

X(\omega ) = \text{ the number of } A_{1}, \ldots , A_{n} \text{ containing } \omega .

Then show that

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(i)

$X$ is a r.v.

(ii)

The distribution of $X$ is $B(n, p)$ (i.e., Binomial with parameters $n$ and $p$ ).

Problem 10.16

Consider the probability space $(\Omega , \mathcal{A}, P)$ , let $\left\{ A_{n}\right\} , n \geq 1$ , be a sequence of events, and set $X_{n} = I_{A_{n}}$ . Then show that the events $\left\{ A_{1}, A_{2}, \ldots \right\}$ are independent if and only if the r.v.s $X_{1}, X_{2}, \ldots$ are independent.

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Problem 10.17

If $X \sim B(n, p)$ , compute the probability that $P$ (number of $H s =$ number of $T s+ r), r = 0,1, \ldots , n$ .

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Problem 10.18

Let $X_{1}, \ldots , X_{n}$ be independent identically distributed (i.i.d.) r.v.s defined on the probability space $(\Omega , \mathcal{A}, P)$ and having d.f. $F$ . Let $F_{n}$ be the empirical d.f. defined in terms of the $X_{i} \mathrm{~ s}$ ; i.e.,

F_{n}(x, \omega ) = \frac{1}{n}\left[\text{ number of } X_{1}(\omega ), \ldots , X_{n}(\omega ) \leq x\right]

Then show that $\sup \left\{ \left|F_{n}(x, \cdot )-F(x)\right| ; x \in \mathbb {R}\right\} \stackrel{\text{ def }}{ = } D_{n}(\cdot )$ is a r.v. That is, although $D_{n}(\cdot )$ is arrived at through noncountable operations, it is still a r.v.

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Problem 10.19

(Glivenko-Cantelli). Refer to Exercise 18 and show that

\sup \left\{ \left|F_{n}(x, \omega )-F(x)\right| ; x \in \mathbb {R}\right\} \underset {n \rightarrow \infty }{\text{ a.s. }} 0.

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