Which of the following properties is correct regarding a cumulative probability function
𝐹
(
𝑥
)
?

Question

Which of the following properties is correct regarding a cumulative probability function 
𝐹
(
𝑥
)
?

BlackTom AI · Accepted Answer

When evaluating a cumulative distribution function F(x) of a random variable, several properties must hold, and each option invites scrutiny of those properties.
Option 1: 'It must be a monotone function.' Indeed, F(x) is required to be non-decreasing: as x increases, the probability that the random variable is less than or equal to x cannot decrease. This monotonicity is a fundamental characteristic of CDFs and ensures that probabilities of events of the form {X ≤ x} behave consistently when x grows.
Option 2: 'A CDF can be greater than 1.' This is false. By definition, P(X ≤ x) is a probability, and probabilities lie in the interval [0, 1]. Therefore F(x) cannot exceed 1 for any x, and values above 1 would violate basic probability axioms.
Option 3: 'We can always find a closed form for the CDF.' This claim is incorrect in general. Many distributions have explicit closed-form expressions for their CDFs (e.g., the normal or exponential), but others are defined implicitly via density functions, simulations, or require numerical methods. Therefore, a universal guarantee of a closed form for every CDF does not hold.
Option 4: 'None of these options are correct.' Given that the first option expresses a true property of CDFs, this statement is false because not all the listed statements are incorrect.
In summary, the only statement that aligns with the essential properties of a cumulative distribution function is the monotonicity condition, while the other statements conflict with the probabilistic foundations or general behavior of CDFs.

Which of the following properties is correct regarding a cumulative probability function 𝐹 ( 𝑥 ) ?Single choice

Similar Questions

The CDF probability for a normal distribution

Question at position 2 To obtain the probability that a continuous random variable falls in a specified range, we find the difference between the cumulative probabilities at the lower and upper end of the:mode.range.median.mean.

Let g(x) be the density function of a continuous random variable \xi. Find the function value of the cumulative distribution function of \xi at 2 if g(x)=\left\{\begin{array}{ccc} 0 & \text{if } & x<1 \\{1\over4} x & \text{if } & 1\le x\le 3\\0 & \text{if } & 3<x \\\end{array}\right..

Which of the following parameters is used to determine the concentration of protein in a spectrophotometer?

What are the units of absorbance?

What is the concentration of protein in the stock solution of BSA used for the standard curve?

When dissolved in water, hydrogen bromide (HBr) forms hydrobromic acid. Determine the hydroxide ion concentration in a 4,500 mL solution containing 3.78 g hydrogen bromide; K = 1.00 x 10¯14.

More Practical Tools for Students Powered by AI Study Helper

Homework AI Solver

Stylized AI Paper Writer

Plagiarism Checker Assistant

Citation AI Academic Writing Tool

In-Class Translation Assistant

AI Note Generator

AI Quiz Answers

Past Exam Questions from University Test Bank

Smart Practice Assistant

Adaptive Practice

Making Your Study Simpler