The Integral of a Random Variable and its Basic Properties

Construct an example where r.v.s $X_{n}$ and $X$, defined on a probability space $(\Omega , \mathcal{A}, P)$, are such that $X_{n} \xrightarrow [n \rightarrow \infty ]{\rightarrow } X$ pointwise (hence $X_{n} \underset {n \rightarrow \infty }{\text{ a.s. }} X$ and $X_{n} \xrightarrow [n \rightarrow \infty ]{P} X$), but $\mathbb {E}\left[X_{n}^{\rightarrow }\right] \underset {n \rightarrow \infty }{\rightarrow } \mathbb {E}\left[X\right]$.

If the r.v. $X \geq 0$ and $\int X d \mu = 0$, then show that $\mu (X \neq 0) = 0$.

If the r.v. $X$ takes on the values $n, n = 1,2, \ldots$, with probability 1 , so that $\sum_{n = 1}^{\infty } \mathbb {P}\left(X = n\right) = 1$, show that $\mathbb {E}\left[X\right] = \sum_{n = 1}^{\infty } \mathbb {P}\left(X \geq n\right)$.

If for the pairwise disjoint events $A_{i}, i = 1, \ldots , n$, the integral $\int_{\sum_{i = 1}^{n} A_{i}} X$ exists, then the integrals $\int_{A_{i}} X, i = 1, \ldots , n$, exist, and $\int_{\sum_{i = 1}^{n} A_{i}} X = \sum_{i = 1}^{n} \int_{A_{i}} X$.

Let $X$ and $Y$ be two simple r.v.s such that $\int X = \int Y = \infty$ or $\int X = \int Y = -\infty$. Then show that $\int (X+Y)$ exists and $\int (X+Y) = \int X+\int Y( = \infty$ or $-\infty )$.

Let $\mu_{1}$ and $\mu_{2}$ be two measures on $(\Omega , \mathcal{A})$ and suppose that $\int X d\left(\mu_{1}+\mu_{2}\right)$ exists. Then $\int X d \mu_{i}$ exist, $i = 1,2$, and $\int X d\left(\mu_{1}+\mu_{2}\right) = \int X d \mu_{1}+\int X d \mu_{2}$.

If the r.v.s $X_{i}, i = 1, \ldots , n$, are integrable, then so is the r.v. $\sum_{i = 1}^{n} X_{i}$ and $\int \sum_{i = 1}^{n} X_{i} = \sum_{i = 1}^{n} \int X_{i}$.

Let $X$ and $Y$ be r.v.s with finite second moments, and set $\mathbb {E}\left[X\right] = \mu_{1}, \mathbb {E}\left[Y\right] = \mu_{2}, 0 < \operatorname {Var}\left[X\right] = \sigma_{1}^{2}, 0 < \operatorname {Var}\left[Y\right] = \sigma_{2}^{2}$. Then the covariance and the correlation coefficient of $X$ and $Y$, denoted respectively, by $\operatorname {Cov}\left[ X, Y \right]$ and $\rho (X, Y)$, are defined by: $\operatorname {Cov}\left[ X, Y \right] = \mathbb {E}\left[\left(X-\mu_{1}\right)\left(Y-\mu_{2}\right)\right] = \mathbb {E}\left[X Y\right]-\mu_{1} \mu_{2}$ and $\rho (X, Y) = \operatorname {Cov}\left[ X, Y \right] / \sigma_{1} \sigma_{2}$.

Let $\Omega = \left\{ -2,-1,3,7\right\}$, and let the measure $\mu$ be defined by: $\mu (\left\{ -2\right\} ) = 2$, $\mu (\left\{ -1\right\} ) = 1, \mu (\left\{ 3\right\} ) = 3, \mu (\left\{ 7\right\} ) = 7$. Let $X$ be a r.v. defined by: $X(-2) = X(-1) = -1$, and $X(3) = X(7) = 1$. Then, for $A = \left\{ -2,3,7\right\}$, compute the integral $\int_{A} X d \mu$.

Let $(\Omega , \mathcal{A}, \mu ) = \left((-5,5), \mathcal{B}_{(-5,5)}, \lambda \right)$ where $\lambda$ is the Lebesgue measure, and let the r.v. $X$ be defined as follows:

Then, for $A = [-1,4]$, compute the integral $\int_{A} X d \mu$.

Let $\Omega = \left\{ 0,1,2, \ldots \right\}$, let $\mathcal{A}$ be the discrete $\sigma$-field of subsets of $\Omega$, and on $\mathcal{A}$, define the function $\mu$ by: $\mu (A) =$ number of nonnegative integers in $A$. Then show that $\mu$ is a measure, and indeed, a $\sigma$-finite measure.

Let $g: \mathbb {R} \rightarrow (0, \infty )$ be nondecreasing in $(0, \infty )$ and symmetric about 0 $(g(-x) = g(x), \quad x \in \mathbb {R})$, and let $X$ be a r.v. such that $\mathbb {E}\left[g(X)\right] < \infty$. Then show that: $\mathbb {P}\left(\left|X\right| \geq c\right) \leq \mathbb {E}\left[g(X)\right] / g(c)$ for every $c > 0$.

For a r.v. $X$, show that

Let $X$ be a simple r.v. defined on $(\Omega , \mathcal{A})$, so that (see Definition 13 in Chapter 1) $X = \sum_{j = 1}^{n} \alpha_{j} I_{A_{j}}$, where $\left\{ A_{j}, j = 1, \ldots , n\right\}$ is a (measurable) partition of $\Omega$, and the $\alpha_{j}, j = 1, \ldots , n$, are assumed to be distinct. Next, let $X = \sum_{i = 1}^{m} \beta_{i} I_{B_{i}}$ be any other representation of $X$. Then show that each $A_{j}, j = 1, \ldots , n$, is the sum of some $B_{i}$ 's.

Remark 6. Under the assumption that the $\alpha_{j}, j = 1, \ldots , n$, are distinct, the partition $\left\{ A_{j}, j = 1, \ldots , n\right\}$ may be called irreducible. For such a partition, there is a unique representation of the respective simple r.v. and consequently, Theorem 1 is rendered superfluous.

If $X$ and $Y$ are two identically distributed integrable r.v.s, then

for any constant $c$.