Which of the following statements correctly verifies that the function $p\left(x\right)=\frac{1}{4}$ for $x=1$, $x=2$, $x=3$, $x=4$ and $p\left(x\right)=0$ otherwise is a probability mass function?

Let the random variables $X$ and $Y$ have joint pdf $f(x,y)=2$ for $0<x<1$, $0<y<x$, and $f(x,y)=0$ otherwise. Find $P(Y<0.5,X>0.5)$ (round off to third decimal place).

Suppose $Z$ is a standard normal random variable. What is the probability that $Z<0$?

Suppose events E and F are such that $P(E\cap F)$ = $P\left(E\right)$ $\times$ $P\left(F\right)$. Are the events E and F independent or dependent? Justify your answer.

Suppose you are given the following data for two variables, $X$ and $Y$: $(1,2)$, $(2,4)$, $(3,6)$, $(4,8)$. What is the value of the sample correlation coefficient between $X$ and $Y$? Do not round your intermediate calculations.

Suppose a normal distribution has mean $\mu$ and standard deviation $\sigma$. According to the empirical rule, what is the approximate probability that a randomly selected value from this distribution falls within one standard deviation of the mean (that is, between $\mu -\sigma$ and $\mu +\sigma$)?

Which of the following terms best describes a process of analyzing data to identify meaningful relations?

If a bag contains only blue and red marbles, and the probability of picking a blue marble is $\frac{3}{25}$, what is the probability of picking a red marble?

If an event has a probability of $\frac{3}{4}$, how likely is it to occur?