## Derivatives and Basic Probability

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

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:

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

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

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