Derivatives and Basic Probability

Here are a few basic exercises from the required prerequisites classes (Calculus I and Discrete Math). I expect you know how to do these, perhaps with a bit of brushing up:

1. Let $$f(x) = 5x^2 + 3$$. Write the derivative of $$f(x)$$.

$$\frac{d}{dx}f(x) = 10x$$

2. Let $$f(x) = 4e^{2x}$$. Write the derivative of $$f(x)$$.

$$\frac{d}{dx}f(x) = 8e^{2x}$$

3. A fair coin has equal probability of coming up heads or tails. If a fair coin is flipped twice, what is the probability of seeing first heads, then tails:

$$\Pr(\text{heads}, \text{tails}) = 1/4$$

4. Consider an unfair coin that has probability $$3/4$$ of coming up heads. If the coin is flipped three times, what is the probability of seeing tails each time:

$$\Pr(\text{tails}, \text{tails}, \text{tails}) = 1/4^3 = 1/64$$

5. I agree to let you flip the same unfair coin one time and pay you $1 if it comes up heads and$2 if it comes up tails. Let $$X$$ be the number of dollars I pay you, which is a random variable. What is the expected value of $$X$$?

$$E[X] = (3/4)\cdot 1 + (1/4) \cdot 2 = 5/4$$

Partial Derivatives and Linear Algebra

These exercises are covered in Multivariate Calculus and Linear Algebra, which are not required prerequisites, but may also be covered in other courses. We will review these topics in class, but it will help if you have seen them before.

1. Let $$f(x,y)= 3xy^2+2y$$. Write the following partial derivative:

$$\frac{\partial}{\partial{x}} f(x,y) = 3y^2$$

2. Let $$f(x,y)= 3xy^2+2y$$. Write the following partial derivative:

$$\frac{\partial}{\partial{y}} f(x,y) = 6xy + 2$$

3. Let $$A$$ be the matrix $$\begin{bmatrix}2 & 3 \\ 5 & 1\end{bmatrix}$$ and let $$\mathbf{x}$$ be the (column) vector $$\begin{bmatrix}1 \\ 2\end{bmatrix}$$. Let $$A^{T}$$ and $$\mathbf{x}^T$$ denote the transpose of $$A$$ and $$\mathbf{x}$$, respectively. Complete the following expressions:

$$A \mathbf{x} = \begin{bmatrix}2 & 3 \\ 5 & 1\end{bmatrix} \cdot \begin{bmatrix}1 \\ 2\end{bmatrix} = \begin{bmatrix}8 \\ 7 \end{bmatrix}$$

$$A^T = \begin{bmatrix}2 & 5 \\ 3 & 1\end{bmatrix}$$

$$\mathbf{x}^T A = \begin{bmatrix}8 & 7 \end{bmatrix}\cdot\begin{bmatrix}2 & 3 \\ 5 & 1\end{bmatrix} = \begin{bmatrix}12 & 5 \end{bmatrix}$$