guide:8a33754ae2: Difference between revisions
No edit summary |
mNo edit summary |
||
Line 126: | Line 126: | ||
\E[X_n]\uparrow\E[X]\text{as $n\to\infty$}. | \E[X_n]\uparrow\E[X]\text{as $n\to\infty$}. | ||
</math> | </math> | ||
<li>(''Fatou'') If <math>(X_n)_{n\geq 1}</math> is a sequence of real valued r.v.'s with <math>X_n\geq 0</math> for all <math>n\geq 1</math>, then | <li>(''Fatou'') If <math>(X_n)_{n\geq 1}</math> is a sequence of real valued r.v.'s with <math>X_n\geq 0</math> for all <math>n\geq 1</math>, then | ||
Line 133: | Line 131: | ||
\E\left[\liminf_{n\to\infty}X_n\right]\leq \liminf_{n\to\infty}\E[X_n]. | \E\left[\liminf_{n\to\infty}X_n\right]\leq \liminf_{n\to\infty}\E[X_n]. | ||
</math> | </math> | ||
<li>(''Dominated convergence'') If <math>(X_n)_{n\geq 1}</math> is a sequence of real valued r.v.'s with <math>\vert X_n\vert \leq Z</math> for all <math>n\geq 1</math>, such that <math>\E[Z] < \infty</math>, for another real valued r.v. <math>Z</math>, and <math>X_n\xrightarrow{n\to\infty}X</math> a.e., then | <li>(''Dominated convergence'') If <math>(X_n)_{n\geq 1}</math> is a sequence of real valued r.v.'s with <math>\vert X_n\vert \leq Z</math> for all <math>n\geq 1</math>, such that <math>\E[Z] < \infty</math>, for another real valued r.v. <math>Z</math>, and <math>X_n\xrightarrow{n\to\infty}X</math> a.e., then | ||
Line 163: | Line 159: | ||
<math>f(X)</math> is also a r.v. | <math>f(X)</math> is also a r.v. | ||
}} | }} | ||
| | |In the case <math>f=\one_B</math> with <math>B\in\mathcal{E}</math> we get that | ||
In the case <math>f=\one_B</math> with <math>B\in\mathcal{E}</math> we get that | |||
<math display="block"> | <math display="block"> | ||
Line 205: | Line 200: | ||
Let <math>d=2</math> and <math>X=(X_1,X_2)</math>. Then <math>P_1(x)=\int_\R P(x,y)dy</math> and <math>P_2(x)=\int_\R P(x,y)dx</math>. | Let <math>d=2</math> and <math>X=(X_1,X_2)</math>. Then <math>P_1(x)=\int_\R P(x,y)dy</math> and <math>P_2(x)=\int_\R P(x,y)dx</math>. | ||
}} | }} | ||
| | |Let <math>\pi_j:(x_1,...,x_d)\mapsto x_j</math>. From Fubini's theorem we get that <math>\forall f:\R\to \R^+</math>, Borel measurable | ||
Let <math>\pi_j:(x_1,...,x_d)\mapsto x_j</math>. From Fubini's theorem we get that <math>\forall f:\R\to \R^+</math>, Borel measurable | |||
<math display="block"> | <math display="block"> |
Latest revision as of 12:32, 8 May 2024
Law of a Random Variable
Let [math](\Omega,\A,\p)[/math] be a probability space. Let [math](E,\mathcal{E})[/math] be a measurable space. A measurable map [math]X:(\Omega,\A,\p)\to (E,\mathcal{E})[/math] is called a random variable (and is noted r.v.) with values in [math]E[/math].
The law or distribution of a random variable is the image measure of [math]\p[/math] by [math]X[/math], and is usually noted [math]\p_X[/math]. It is hence a probability measure on [math](E,\mathcal{E})[/math].
If [math]\mu[/math] is a probability measure on [math](\R^d,\B(\R^d))[/math], (or even on a more general space [math](E,\mathcal{E})[/math]), there is a canonical way of constructing a r.v. [math]X[/math] such that [math]\p_X=\mu[/math] as a map
There are two special cases.
- Discrete r.v.: Let [math]E[/math] be a countable space and [math]\mathcal{E}=\mathcal{P}(E)[/math]. The law of [math]X[/math] is given by
[[math]] \p_X:=\sum_{x\in E}P(x)\delta_x, [[/math]]where [math]P(x)=\p[X=x][/math] and [math]\delta_x[/math] is the Dirac measure of [math]x[/math], meaning that for all [math]A\subset E[/math],[[math]] \delta_x(A)=\begin{cases}1&\text{if $x\in A$}\\ 0&\text{if $x\not\in A$}\end{cases} [[/math]]We note that if [math]\p_X[E]=1[/math], then[[math]] \sum_{x\in E}P(x)\delta_x(E)=\sum_{x\in E}P(x)=1. [[/math]]Indeed, for all [math]B\in E[/math] we have that[[math]] \p_X[B]=\p[X\in B]=\p\left[\bigcup_{x\in B}\{X=x\}\right]=\sum_{x\in B}\p[X=x]=\sum_{x\in E}P(x)\delta_x(B). [[/math]]
- Continuous r.v.: A random variable [math]X[/math] with values in [math](\R^d,\B(\R^d))[/math] is said to have a density if [math]\p_X\ll \lambda[/math], where [math]\lambda[/math] is the lebesgue measure on [math]\R^d[/math]. The Radon-Nikodym theorem says there exists [math]P:\R^d\to\R[/math], measurable such that for al [math]B\in \B(\R^d)[/math]
[[math]] \p_X[B]=\int_BP(x)dx. [[/math]]In particular, [math]\int_{\R^d}P(x)dx=\p_X(\R^d)=1[/math]. Moreover the map [math]P[/math] is unique up to sets of lebesgue measure 0. [math]P[/math] is called the density of [math]X[/math]. If [math]d=1[/math], then[[math]] \p[\alpha\leq X\leq \beta]=\p_X[[\alpha,\beta]]=\int_\alpha^\beta P(x)dx. [[/math]]
Let [math](\Omega,\A,\p)[/math] be a probability space. Let [math]X[/math] be a real valued r.v. (i.e. with values in [math]\R[/math]). The expectation of such a r.v. is defined as
- If [math]x\geq 0[/math], and then [math]\E[X]\in[0,\infty][/math].
- If [math]\E[X]=\int_{\Omega}\vert X(\omega)\vert d\p(\omega) \lt \infty[/math].
We extend this definition to the case of a r.v. [math]X=(X_1,...,X_d)[/math] taking values in [math]\R^d[/math] by defining
provided each [math]\E[X_i][/math] is well defined.
If [math]B\in\A[/math] and [math]X=\one_{B}[/math], then
The expectation is a special case of an integral with respect to a positive measure. In particular,
- For all [math]X,Y[/math] integrable and [math]a,b\in\R[/math] we have
[[math]] \E[aX+bY]=a\E[X]+b\E[Y]. [[/math]]
- If [math]C[/math] is a constant and [math]\E[X]=C[/math], then
[[math]] \int_\Omega Cd\p(\omega)=C\p[\Omega]=C. [[/math]]
- If [math]X\geq 0[/math] and [math]\E[X]\geq 0[/math] and if [math]X\leq Y[/math] both integrable then
[[math]] \E[X]\leq \E[Y]. [[/math]]
- (Monotne convergence) If [math](X_n)_{n\geq 1}[/math] is a sequence of real valued r.v.'s, and if [math]X_n\geq 0[/math] for all [math]n\geq 1[/math] and [math]X_n\uparrow X[/math] as [math]n\to\infty[/math], then
[[math]] \E[X_n]\uparrow\E[X]\text{as $n\to\infty$}. [[/math]]
- (Fatou) If [math](X_n)_{n\geq 1}[/math] is a sequence of real valued r.v.'s with [math]X_n\geq 0[/math] for all [math]n\geq 1[/math], then
[[math]] \E\left[\liminf_{n\to\infty}X_n\right]\leq \liminf_{n\to\infty}\E[X_n]. [[/math]]
- (Dominated convergence) If [math](X_n)_{n\geq 1}[/math] is a sequence of real valued r.v.'s with [math]\vert X_n\vert \leq Z[/math] for all [math]n\geq 1[/math], such that [math]\E[Z] \lt \infty[/math], for another real valued r.v. [math]Z[/math], and [math]X_n\xrightarrow{n\to\infty}X[/math] a.e., then
[[math]] \E[X_n]\xrightarrow{n\to\infty}\E[X]. [[/math]]
In probability theory we say almost sure convergence and write a.s., rather than almost everywhere. If [math]X_n\xrightarrow{n\to\infty}X[/math] a.s., then we mean
Let [math]X[/math] be a r.v. with values in [math](E,\mathcal{E})[/math]. If [math]f:E\to [0,\infty][/math] is measurable, then
[math]f(X)[/math] is also a r.v.
In the case [math]f=\one_B[/math] with [math]B\in\mathcal{E}[/math] we get that
One often uses the proposition to compute the law of a r.v. [math]X[/math]. If one is able to write [math]\E[X]=\int f d\nu[/math] for a sufficiently large class of functions [math]f[/math], then one can deduce that [math]\p_X=\nu[/math]. The idea is to be able to take [math]f=\one_B[/math], for then [math]\E[f(X)]=\p_X[B]=\nu(B)[/math].
Example
Assume that [math]\p_X[/math] is absolutely continuous with density [math]h(x)=\frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}}[/math] for [math]x\in\R[/math] and [math]Y=X^2[/math]. Then one can ask about the distribution of [math]Y[/math]. Let [math]f:\R\to[0,\infty][/math] be measurable. Then
We can write
Now we can set [math]y=x^2[/math]. Then [math]dy=2xdx[/math] and hence [math]dx=\frac{dy}{2\sqrt{y}}[/math]. Now we can write
which implies that
So we see that the distribution of [math]Y[/math] is given by [math]\frac{e^{-\frac{y}{2}}}{\sqrt{2\pi y}}\one_{\{y \gt 0\}}[/math].
Let [math]X=(X_1,...,X_d)\in\R^d[/math] be a r.v. Assume that [math]X[/math] has density [math]P(x_1,...,x_d).[/math] Then [math]\forall j\in\{1,...,n\}[/math], [math]X_j[/math] has density
Let [math]d=2[/math] and [math]X=(X_1,X_2)[/math]. Then [math]P_1(x)=\int_\R P(x,y)dy[/math] and [math]P_2(x)=\int_\R P(x,y)dx[/math].
Let [math]\pi_j:(x_1,...,x_d)\mapsto x_j[/math]. From Fubini's theorem we get that [math]\forall f:\R\to \R^+[/math], Borel measurable
If [math]X=(X_1,...,X_d)\in\R^d[/math] is a r.v., then the distribution [math]\p_{X_j}[/math] are called the margins of [math]X[/math]. The last proposition shows us that the margins are determined by
General references
Moshayedi, Nima (2020). "Lectures on Probability Theory". arXiv:2010.16280 [math.PR].