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:

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

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

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