guide:4674549cf7: Difference between revisions
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A random vector <math>Z=(Z_1,...,Z_n)</math> is said to be Gaussian, if for all <math>\lambda_1,...,\lambda_n\in\R</math> | A random vector <math>Z=(Z_1,...,Z_n)</math> is said to be Gaussian, if for all <math>\lambda_1,...,\lambda_n\in\R</math> | ||
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span\{Y_1,...,Y_p\}. | span\{Y_1,...,Y_p\}. | ||
</math> | </math> | ||
Consequently, there exists real numbers <math>\lambda_1,...,\lambda_p</math> such that | Consequently, there exists real numbers <math>\lambda_1,...,\lambda_p</math> such that | ||
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\E[X\mid Y_1,...,Y_p]=\lambda_1 Y_1+\dotsm +\lambda_p Y_p. | \E[X\mid Y_1,...,Y_p]=\lambda_1 Y_1+\dotsm +\lambda_p Y_p. | ||
</math> | </math> | ||
{{alert-info | Moreover, for a measurable map <math>h:\R\to\R_+</math> we get | |||
<math display="block"> | <math display="block"> | ||
\E[h(X)\mid Y_1,...,Y_p]=\int_\R h(x)Q_{\sum_{j=1}^n\lambda_j Y_j,\sigma^2}(x)dx, | \E[h(X)\mid Y_1,...,Y_p]=\int_\R h(x)Q_{\sum_{j=1}^n\lambda_j Y_j,\sigma^2}(x)dx, | ||
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Q_{n,\sigma^2}(x)=\frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{(x-m)^2}{2\sigma^2}\right). | Q_{n,\sigma^2}(x)=\frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{(x-m)^2}{2\sigma^2}\right). | ||
</math> | </math> | ||
Proof: Exercise.{{efn|This is done similarly to the proof of [[#thm1 |theorem]]}}}}|Let <math>\tilde X=\lambda_1 Y_1+\dotsm +\lambda_pY_p</math> be the orthogonal projection of <math>X</math> onto <math>span\{Y_1,...,Y_p\}</math>, meaning that for all <math>1\leq j\leq p</math> | |||
Exercise.{{efn|This is done similarly to the proof of [[#thm1 |theorem]]}}}} | |||
Let <math>\tilde X=\lambda_1 Y_1+\dotsm +\lambda_pY_p</math> be the orthogonal projection of <math>X</math> onto <math>span\{Y_1,...,Y_p\}</math>, meaning that for all <math>1\leq j\leq p</math> | |||
<math display="block"> | <math display="block"> | ||
\E[(X-\tilde X)Y_j]=0. | \E[(X-\tilde X)Y_j]=0. | ||
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<math display="block"> | <math display="block"> | ||
\E[X\mid Y_1,...,Y_p]=\E[X-\tilde X+\tilde X\mid Y_1,...,Y_p]=\E[X-\tilde X\mid Y_1,...,Y_p]+\E[\tilde X\mid Y_1,...,Y_p]=\E[X-\tilde X]+\tilde X=\tilde X. | \E[X\mid Y_1,...,Y_p]=\E[X-\tilde X+\tilde X\mid Y_1,...,Y_p]=\E[X-\tilde X\mid Y_1,...,Y_p]+\E[\tilde X\mid Y_1,...,Y_p]=\E[X-\tilde X]+\tilde X=\tilde X. | ||
</math> | </math>}} | ||
==General references== | ==General references== | ||
{{cite arXiv|last=Moshayedi|first=Nima|year=2020|title=Lectures on Probability Theory|eprint=2010.16280|class=math.PR}} | {{cite arXiv|last=Moshayedi|first=Nima|year=2020|title=Lectures on Probability Theory|eprint=2010.16280|class=math.PR}} | ||
==Notes== | ==Notes== | ||
{{notelist}} | {{notelist}} |
Latest revision as of 21:28, 8 May 2024
A random vector [math]Z=(Z_1,...,Z_n)[/math] is said to be Gaussian, if for all [math]\lambda_1,...,\lambda_n\in\R[/math]
is Gaussian.Moreover, [math]Z[/math] is called centered, if [math]\E[Z_j]=0[/math] for all [math]1\leq j\leq n[/math]. Let [math]Z[/math] be a Gaussian vector. Then for all [math]\xi\in\R^n[/math] we get
where [math]C_Z:=(C_{ij})[/math] and [math]C_{ij}=\E[Z_iZ_j][/math]. If [math]Cov(Z_i,Z_j)=0[/math], then [math]Z_i[/math] and [math]Z_j[/math] are independent. More generally, we have the Gaussian vectors
[math]Y_1[/math] and [math]Y_2[/math] are independent if and only if [math]Cov(X_j,X_n)=0[/math], where [math]1\leq j\leq i_1[/math] and [math]i_1+1\leq k\leq i_2[/math]. If [math]Z_1,...,Z_n[/math] are independent Gaussian r.v.'s, we have that
is a Gaussian vector. If [math]Z[/math] is a Gaussian vector and [math]A\in \mathcal{M}(m\times n,\R)[/math] , we get that [math]AZ[/math] is again a Gaussian vector.
Let [math](\Omega,\F,\p)[/math] be a probability space. Let [math]X\in L^1(\Omega,\F,\p)[/math] and [math]Y_1,...,Y_p\in L^1(\Omega,\F,\p)[/math] and let [math](X,Y_1,...,Y_p)[/math] be a centered Gaussian vector. Then
Consequently, there exists real numbers [math]\lambda_1,...,\lambda_p[/math] such that
Let [math]\tilde X=\lambda_1 Y_1+\dotsm +\lambda_pY_p[/math] be the orthogonal projection of [math]X[/math] onto [math]span\{Y_1,...,Y_p\}[/math], meaning that for all [math]1\leq j\leq p[/math]
General references
Moshayedi, Nima (2020). "Lectures on Probability Theory". arXiv:2010.16280 [math.PR].
Notes
- This is done similarly to the proof of theorem