Lecture 2025-01-06
Random vectors in $\mathbb{R}^d, d \geq 2$. \(X=\left(\begin{array}{c} x_1 \\ \vdots \\ x_0 \end{array}\right)\)
\[P\left(X \in \sum_{i=1}^j\left[a_i, b_i\right] .\right)\] \[\bar{X}_1 \in\left[a_1, b_1\right], x_2 \in\left[a_2, b_2\right], \ldots, x_d \in\left[a_d, b_d\right]\]Examples Binomial 2 outcomes. this how more than 2. (1) $Z \sim \operatorname{multinomial}\left(n ; p_1, \ldots, p_d\right)$ where $\sum_{i=1}^j p_i=1 p_1 \geq 0$ $d$ outcomes: $O_1, \ldots, O_j$
$X = \left(X_i\right)_{i \leq i \leq d} X_i=$ # of outcomes $i \quad 1 \leq i \leq d$.
\[X_1 \in(0,1, \ldots, n)\]$X=\sum_{i=1}^n Y_i, \quad\left{Y_i\right}_{1 \leq k \leq d}$ Ind.
\[Y_k \in\left\{\left(\begin{array}{l} 0 \\ \vdots \\ 0 \\ 1 \\ 0 \\ \vdots \\ 0 \end{array}\right)-X_i\right\} \text { if } i \text { occured in } k \text { experiment. }\]$\therefore x_1$ is … $x_1 \sim$ Binomial $\left(n, p_1\right)$
\[P\left(Z=\left(\begin{array}{c} n_1 \\ \vdots \\ n_0 \end{array}\right)\right)=\frac{n!}{\prod_{k=1}^n (n k)!} \prod_{k=1}^{d} P_i^{n_i}\left\{\begin{array}{l} \text { CDF } \\ \text { of } \\ \text { Multinomial } \end{array}\right. \\ \sum_{c=1}^{d} n_i=n .\]People are interested in coverince matrix of $X$. have $d$ coordinates $d \times d$ matrix. Covariance matrix $\Gamma(X)=\left[\Gamma_{i j}\right]_{1 \leqslant i, j \leqslant d}$.
\(\begin{aligned} \Gamma_{i, j} & =\operatorname{Cov}\left(X_i, X_j\right) \\ & =E\left(X_i, X_j\right)-E\left(X_i\right) E\left(X_j\right) \\ \Gamma_{i, i} & =V\left(X_i^{\prime}\right)=1 p_i\left(1-p_i\right) \quad 1 \leq i \leq \partial . \\ \Gamma_{i, j} & =E\left(X_i, X_j\right)-E\left(X_i\right) E\left(X_j\right) \end{aligned}\) — \(\begin{aligned} & X_i=\sum_{k=1}^n \varepsilon_k, \varepsilon_k \sim \operatorname{Ber}\left(P_i\right), \quad\left\{\varepsilon_k\right\}_{1 \leq k \leq n} \text { ind. } \\ & X_j=\sum_{l=1}^n \delta_l, \quad \delta_l \sim \operatorname{Ber}\left(P_j\right) \quad\left\{\delta_l\right\}_{1 \leq k s n \text { ind. }} \\ & \operatorname{Cov}\left(X_i, X_i\right)=\sum_{1 \leq k, l \leq n} \operatorname{Cov}\left(\varepsilon_k, \delta_l\right)=\sum_{k=1}^n \operatorname{Cov}\left(\varepsilon_k, \delta_k\right) \end{aligned}\)
remark. Either $\varepsilon_1 = 1$ and $d_1 =0$, $\varepsilon_1=0, d_1 = 1$, $\therefore e_1d_1=0$.
\[\begin{aligned} =n \operatorname{cov}\left(\varepsilon_1 \delta_1\right)= & n\left[E\left(\varepsilon_1 \delta_1\right)-E\left(\varepsilon_1\right) E\left(\delta_1\right)\right] \\ & n\left[O-p_i p_j \quad\right. \text { cov } \\ \therefore \quad & \Gamma_{i, j}= - np_ip_j \end{aligned} **remark**. The coveriance is negative.\]$X \sim$ multivariate normal $(\mu, \Gamma) \quad \mu \in \mathbb{R}^d$
$\vec{\mathbb{E}(X)}=\vec{\mu}$
\[E(x)=\left(\begin{array}{c} \mathbb{E}x_n\\ \vdots \\ \mathbb{E}X_n \end{array}\right)\]$\Gamma=\left[\operatorname{cov}\left(X_i, X_j\right)\right]_{1 \leqslant i, j \leqslant d}$
if $\vec{X}^\mathcal{D}=\vec{M}+A \vec{Z}$ Where $A \in M_{d \alpha d} \leftarrow$ matrix
$A=\left[a_{i j}\right]_{1 \leq i, j \leq d}$
\[\vec{Z}=\left(\begin{array} { c } Z_1 \\ \vdots\\ Z_d \end{array} \right)\] \[\{Z_i\}_{1\le i\le d} \text{ ind } N(0,1)\]what happend to covariance matrix
$\vec{Y} \in \mathbb{R}^d, E(Y)=0$
$\Gamma(Y)=YY^T$
$\mathbb{R}^{\delta \times \delta}$
$ \Gamma(Y)=E\left(Y \cdot Y^{+}\right)$
$ \Gamma=E\left((x-\mu)(x-\mu)^T\right)$
$\qquad=A Z(A Z)^T$
remark. $(A B)^t=B^t A^t$
$=A Z Z^t A^t$
$A E \underbrace{\left(z z^t\right)}_I A^t$
$=A A^t$
$A A^t$ is symmetric matrix.
X is R.V. is Multinormal iff $t \cdot X=\langle t, X\rangle$ is $N\left(\mu, \sigma^2\right) . \quad \forall \vec(t) \in \mathbb{R}^d$. $t X=\sum_{i=1}^n t_i x_i$
Multivariate and multinormalss are 2 important examples how to characterize the distribution.
Let $X \in \mathbb{R}^{+}$ \(F_x\left(t_1, \ldots, t_d\right)=P\left(x_1 \leq t_1, \ldots, x_d \leq t_d\right), \quad Z \in \mathbb{R}^2\) you charaterize to the distribution with rectangles
The Important char is not that even through books says Disitribution of X is characterized by $\operatorname{Dis}{t\cdot X}_{t\in\mathbb{R}^d}.
Namely, $P(t \cdot X \leq u) \quad u \in \mathbb{R}$. this looks like half plane ${tX\le u}$ in 2d.
if $|t|2=1 \quad$ Norm $\quad \sum{t=1}^2 t_i^2=1$ since for all $t$ we can consider $u$ part of $t$, \(\varphi_X(t)=\varphi_{t \cdot x}(1)=E e^{i(t \cdot X)}, t \in \mathbb{R}^d.\)
Claim this is the characteristic function. of $x$. \(X \in \mathbb{R}^{d}, t \in \mathbb{R}^{d}\) inversion formala Proved similiar to W. no cher. Func in infinite Dim. Problem use, $\varphi_x(t)$ to calculute $P(x \in A)$ where \(\begin{aligned} & A=\sum_{i=1}^d\left(a_i, b_i\right) \\ & P\left(X \in J_A\right)^K \text { the boundry. } \end{aligned}\) $x$ subtract from ind. uniform. \(\begin{array}{ll} Y=X-U & U \sim \text { uniform }(A) \\ & f_U(t)=\frac{1}{\text { volue }(A)} \quad X \in A . \end{array}\)
Where $U \text{ and } x$ ind.
if $x \notin A$ then $0$.
$Y$ is Bounded, intrgable, Even of $x$ is next, \(f_{\vec{y}}(\overrightarrow{0})=P(x \in A) .\)
Lecture 2025-01-15
Last time inverson formula. $$ X \in \mathbb{R}^d \varphi_{X}(t)=E e^{i(t-X)} t \in \mathbb{R}^d
t X=\sum_{i=1}^d t_i X_i $$
\(A=\prod_{i=1}^A\left[a_i b_i\right]\) Invession formla move from $U$ to $T$ because $\varphi_U(t)=E e^{i t U}$. \(P(x \in A)=\lim _{T \rightarrow \infty}(2 \pi)^{-\alpha} \lambda(A) \cdot \int_{[-T, T]^d} \varphi_U(-t) \varphi_X(t) dt\) Lebesque Measure Aka Volume
Important trick.
$$
Y=X-U, \quad U \sim \text { uniform }(A), \quad(X, U) \text { InD. }
\varphi_Y(t)=E\left(e^{i t \cdot(X-U)}\right)=E\left(e^{i t x} \cdot e^{i(-t) U}\right)
f_Y(0)=\frac{P(x \in A)}{\lambda(A)}=\varphi_x(t) \cdot \varphi_U(-t) .
$$
Let $Y$ be Random vector. with “nice” density and $\varphi_Y(t)$ is the ch.f. then $\lim {T \rightarrow \infty}(2 \pi)^{-\delta} \int{[-T, T]^d} \varphi_Y(t) dt =f_Y(0)$
Where [-T, T]^d represents an expanding hyper cube.
If we use fubini mass accummulates at 0.
$$
f_U(v)=\frac{1}{\lambda(A)}, \quad v \in A
v=\left(\begin{array}{c}
u_1
\vdots
u_d
\end{array}\right)
U_k \sim \text { uniform }\left(a_k, b_k\right)
\perp!!!\perp\left(U_k\right)_{1 \leq k \leq 0}
$$
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