Cumulative Distribution Function (CDF)

Which of the following statements accurately describes the cumulative distribution function (CDF)?

A random variable $Y$ has a CDF given by $F_Y(y) = 0$ for $y < 0$, $F_Y(y) = 0.2y$ for $0 \leq y \leq 5$, and $F_Y(y) = 1$ for $y > 5$. What is the probability that $Y$ is less than or equal to 3?

If $F_X(x)$ is the CDF of a random variable $X$, which of the following properties is always true?

A fair six-sided die is rolled. Let $X$ be the number shown on the die. What is the value of the CDF, $F_X(x)$, at $x = 3.5$?

Suppose the CDF of a random variable $X$ is given by $F_X(x)$. How can you find the probability that $X$ lies in the interval $(a, b]$, where $a < b$?

Which of the following is NOT a property of a cumulative distribution function (CDF)?

Let $X$ be a discrete random variable with the following probability mass function (PMF): $P(X = -1) = 0.2$, $P(X = 0) = 0.5$, and $P(X = 1) = 0.3$. What is the value of the cumulative distribution function (CDF) at $x = 0.5$, i.e., $F_X(0.5)$?

The CDF, $F_X(x)$, for a random variable $X$ is given. Which expression represents the probability that $X$ is strictly greater than $a$?

Two coins are tossed. Let $X$ be the number of tails. What is the CDF, $F_X(x)$, evaluated at $x=1.5$?

Which of the following is true regarding the relationship between the Probability Mass Function (PMF) and the Cumulative Distribution Function (CDF) for a discrete random variable X?

A random variable $X$ has the following CDF: $F_X(x) = 0$ for $x < 1$, $F_X(x) = 0.4$ for $1 \leq x < 3$, $F_X(x) = 0.8$ for $3 \leq x < 5$, and $F_X(x) = 1$ for $x \geq 5$. What is $P(X=3)$?

If you are given the CDF of a random variable, how do you determine if the random variable is continuous?

The cumulative distribution function (CDF) of a random variable X is given by $F_X(x) = \begin{cases} 0, & x < 0 \ x^2, & 0 \leq x < 1 \ 1, & x \geq 1 \end{cases}$. What is the probability that X is between 0.5 and 0.75, i.e., $P(0.5 < X \leq 0.75)$?

Which of the following is a key difference between a PMF and a CDF?

Suppose $X$ is a continuous random variable with CDF $F_X(x)$. What does $F_X'(x)$ represent?

Given a random variable $X$, the value of $F_X(-\infty)$ is always equal to:

For a discrete random variable, how do you calculate $P(a < X \leq b)$ using the CDF $F_X(x)$?

Is it possible to have a CDF, $F_X(x)$, where $F_X(5) = 1.5$?

Let $X$ be a random variable. If $F_X(x)$ is the CDF of $X$, then $\lim_{x \to \infty} F_X(x)$ is equal to:

Suppose you have the CDF for a random variable $X$. Which of the following operations would NOT be valid?

A random variable $X$ has a CDF given by $F_X(x) = cx^2$ for $0 \le x \le 2$, and $F_X(x) = 0$ for $x < 0$ and $F_X(x) = 1$ for $x > 2$. What is the value of the constant $c$?

What does a flat (horizontal) region in the graph of a CDF indicate?

Suppose a random variable $X$ only takes non-negative integer values. Which of the following statements must be true about its CDF $F_X(x)$?

Which statement correctly describes the relationship between the CDF and statistical inference?

Let $X$ be a random variable with CDF $F_X(x)$. What is $P(X < a)$ in terms of $F_X(x)$?

A gambler plays a game where the probability of winning is 0.6. Let $X$ be 1 if the gambler wins and 0 if they lose. What is the CDF, $F_X(x)$, evaluated at $x=0.5$?

If $F_X(x)$ and $F_Y(y)$ are CDFs of random variables $X$ and $Y$ respectively, and $F_X(z) > F_Y(z)$ for some value $z$, what does this imply?

Which of the following statements best describes the behavior of any CDF as $x$ approaches negative infinity?