Definite Matrices

Can we Express an Indefinite Matrix as Sum of Positive Definite/Semi Definite and Negative Definite/Semi Definite matrices? I am working on it...

Conditional Gaussian Distribution

Let $X_1$ and $X_2$ are Jointly Gaussian Random Variables(Which Implies They are Individually Gaussian) i.e.,

$$ X_1 \sim \mathcal{N}(0,\sigma_1^2)$$ and
$$X_2 \sim \mathcal{N}(0,\sigma_2^2)$$, with Correlation Coefficient as $\rho$.

Then we Know that :

$$f_{X_1X_2}(x_1,x_2)=\\ \\ \left(\frac{1}{2 \pi \sigma_1 \sigma_2 \sqrt{1-\rho^2}} \right)\times
\mathbf{e^{\frac{-1}{2(1-\rho^2)}\left(\frac{x_1^2}{\sigma_1^2}-\frac{2 \rho x_1 x_2}{\sigma_1 \sigma_2}+\frac{x_2^2}{\sigma_2^2} \right )}}\\ \\ \;\;\;\; \forall\; x_1,x_2 \in (-\infty\; \infty)\\
\\ $$

Prove that :
$$ E(X_2|X_1)= \frac{\rho x_1 \sigma_2}{\sigma_1} $$
$$ Var(X_2|X_1)=\sigma_2^2\,(1-\rho^2) $$

Find the Digit

This is a Pretty Logical Question..

1216451*0408832000 is 19!, ! Denotes Factorial.  Find the Digit in the place of *

Unitary and Orthonormal Matrix

Let the Matrix $U \in \mathbb{C}^{N \times N}$ is Unitary. Define $\widetilde{U} \in \mathbb{R}^{2N \times 2N}$ as:

$$ \widetilde{U}=\begin{bmatrix}
Re(U) & -Im(U)\\
\\
 Im(U)& Re(U)
\end{bmatrix}$$

Prove that $\widetilde{U}$ is Orthonormal

Complex Gaussian Random Vector

Consider an $N$ Dimensional Complex Gaussian Random Vector $Z \in \mathbb{C}^N$, Such That

$$ Z=X+jY $$, where $X \in \mathbb{R}^N$ and $ Y \in \mathbb{R}^N$ are $N$ Dimensional Real Gaussian Random Vectors .

Covariance matrix $Q$ of $Z$ in terms of Auto Covariance and Cross Covariance Matrices of $X$ and $Y$ is Defined as:

$$ Q=E((Z-E(Z))(Z-E(Z))^H)$$

So:
$$ Q=(\Sigma_{XX}+\Sigma_{YY})-j(\Sigma_{XY}-\Sigma_{XY}^T)$$

Define $2N$ Dimensional Gaussian Random Vector $\widetilde{Z}$, as:

$$ \widetilde{Z}= \begin{bmatrix}
X\\
Y
\end{bmatrix}$$

The Covariance Matrix of $\widetilde{Z}$ is Denoted as $\widetilde{K}$, Defined as:

$$ \widetilde{K}=E((\widetilde{Z}-E(\widetilde{Z}))(\widetilde{Z}-E(\widetilde{Z}))^T)$$

So $\widetilde{K}$, In terms of Auto Covariance and Cross Covariance Matrices of $X$ and $Y$ is:

$$ \widetilde{K}=\begin{bmatrix}
\Sigma_{XX} & \Sigma_{XY} \\
\\
\Sigma_{XY}^T & \Sigma_{YY}
\end{bmatrix}$$

Now if $Z$ is Circularly Symmetric, Prove That:

$$ \widetilde{K}=\frac{1}{2}\begin{bmatrix}
Re(Q) & -Im(Q)\\
\\
Im(Q) & Re(Q)
\end{bmatrix} $$

Affine Transformation Does not Alter Correlation Coefficient

if $X$ and $Y$ are Two Random Variables with Correlation Coefficient $\rho$, Then the Correlation Coefficient of Random Variables
$$ a_1X+b_1\;\text{and}\; a_2Y+b_2$$, Where $a_1$ and $a_2$ are Non Zero Real Numbers with Same Sign is also $\rho$. if They have Opposite Signs, The Correlation Coefficient is $-\rho$.

Laplacian Distribution

Consider the Laplacian Distribution whose PDF is Given by:

$$ f_X(x)=\frac{1}{2}\,e^{-|x|}\,\, {-\infty}<x<{\infty} $$

If Random Variable $Y$ is Defined as

$$ Y=|X|+|X-3| $$,  Find

$$ Pr(Y\geq 3) $$