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

$$Pr(Y\geq 3)$$