DRW OA Mixture | Yiming Zhang

Null Space of A Matrix

What is the dimension of the null space of the matrix $$ \left[\begin{array}{ccccc} 1 & 3 & 3 & 4 & 5 \\ 0 & 2 & 4 & 6 & 10 \\ 2 & 6 & 6 & 8 & 10 \\ 0 & 0 & 5 & 2 & 7 \\ 0 & 3 & 6 & 9 & 15 \end{array}\right] $$

The column and row rank are always the same. In the case, by eliminating the two rows that are multiplier of the others, we get the rank of the matrix is $3$. The null space has a dimension of $5-3=2$.

Probability

Let $X$ be a random variable on $\mathbb{R}$ such that $P[X<1]=0, P[X>e]=0$, and for $y \in[1, e], P[X < y]=\int_{1}^{y} \frac{1}{x} d x .$ Let $\mu$ denote the mean value of $X$. Which integer is closest to $10 k$, where $k$ is the value such that $P[X-\mu>k]=0.05$ ?

This solution is in a written format. Please find it at the last section of this page.

Gaussian Noise to Regularize Parameters

Adding noise to the predictors is equivalent to conduct ridge regression: https://stats.stackexchange.com/questions/548756/how-is-adding-noise-to-training-data-equivalent-to-regularization

Linear Regression - Heteroskedastic

Average Time Between Customers to Supervisor

Trace of Matrix Exponential

Let $T$ be the trace of $e^{A}$, where $A$ is the $2 \times 2$ matrix $\left[\begin{array}{cc}2 & 3 / 2 \\ -1 & -1 / 2\end{array}\right]$. Which integer is CLOSEST to the value of $3 T$ ?

Reference: https://math.stackexchange.com/questions/1821508/trace-of-matrix-exponential-closed-form-expression

Conclusion is, $$ \operatorname{Tr}(\exp (A))=\sum_{\lambda \in \Lambda(A)} m_{\lambda} e^{\lambda} $$ where $\Lambda(A)$ is the set of distinct generalized eigenvalues of $A$, and $m_{\lambda}$ is the multiplicity corresponding to $\lambda \in \Lambda(A)$.

For this problem, we can find for matrix $A$, the two eigenvalues are $1/2$ and $1$.So trace of $e^A$ is $T =e^{0.5}+e^1=4.3670$. Then $3T=13.101$.