Code snippet Consider the set $S$ of points $(x, y) \in \mathbb{R}^2$ which minimize the real valued function $f(x, y) = (x + y - 1)^2 + (x + y)^2$ Which of the following statements is true about the set $S$?

Let $X$ and $Y$ be continuous random variables with probability density functions $PX(x)$ and $PY(y)$, respectively. Further, let $Y = X^2$ and $PX(x) = \begin{cases} 1, & x \in (0,1 \\ 0, & \text{otherwise} \end{cases}$ Which one of the following options is correct?

Consider discrete random variables $X$ and $Y$ with probabilities as follows: $P(X = 0 \text{ and } Y = 0) = \frac{1}{4}$ $P(X = 1 \text{ and } Y = 0) = \frac{1}{8}$ $P(X = 0 \text{ and } Y = 1) = \frac{1}{2}$ $P(X = 1 \text{ and } Y = 1) = \frac{1}{8}$ Given $X = 1$, the expected value of $Y$ is

Let $A = \begin{bmatrix} 1 & 1 & 1 \\ -1 & -1 & -1 \\ 0 & 1 & -1 \end{bmatrix}$, and $b = \begin{bmatrix} 1/3 \\ -1/3 \\ 0 \end{bmatrix}$. Then, the system of linear equations $Ax = b$ has

Let $P = \begin{bmatrix} 2 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}$ and let $I$ be the identity matrix. Then $P^2$ is equal to