Homework 0

This math assessment is meant to help both you and the instructor identify gaps in background knowledge both at the class and individual level.

Important

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 β, 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?