Homework 0

The purpose of this math assessment is to help both you and the instructor identify gaps in background knowledge both at the class and individual level. For this reason, do not use external resources to complete this assignment. Even though you will submit to Gradescope, it is not for a grade. If you don’t know the answer, just leave it blank.

Important

Again, this homework will not be a part of your grade.

Exercise 1

Simplify

\[ \log(e^{a_1} e^{a_2} e^{a_3} \cdots e^{a_n}) \]

Exercise 2

Find the derivative,

\[ \frac{d}{dx} \left( \frac{x}{\log x} \right) \]

Exercise 3

What is the ordinary least squares estimator of \(\beta\) (1-dimensional) in the linear regression \(y = x \beta + \epsilon\) with iid errors?

Exercise 4

What is the ordinary least squares estimator of \(\beta\) (p-dimensional) in the linear regression \(y = X \beta + \epsilon\) with iid errors?

Exercise 5

In linear regression with p-dimensional \(\beta\), what is the interpretation of the estimate for the jth coefficient?

Exercise 6

Compute the integral,

\[ \int_{-\infty}^{\infty} e^{-x^2} dx \]

Exercise 7

\(X \sim N(\mu, \sigma^2)\) reads “X is normally distributed with mean \(\mu\) and variance \(\sigma^2\).

Let

\[ \begin{aligned} X &\sim N(0, 1),\\ Y &\sim N(3, 2),\\ Z &= X + Y \end{aligned} \] What is the distribution of \(Z\)? What is \(\mathbb{E}[Z]\) and \(Var(Z)\)?

Exercise 8

In your own words, the “support” of random variable is…

Exercise 9

TRUE/FALSE: The product of two uniform[0, 1] random variables is uniform[0, 1].

Exercise 10

\(X_1, \ldots X_n\) i.i.d. with pdf \(p(x)\). For all \(i\), \(E(X_i) = \mu\) and \(Var(X_i) = \sigma^2 < \infty\).

Compare \(\text{Var}(\frac{1}{n}\sum X_i)\) to \(\text{Var}(X_1)\). Which is smaller?