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.

25F-STATS-102C-LEC-1 Test Portal for Lockdown Browser- Requires Respondus LockDown Browser

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

查看解析

登录即可查看完整答案

类似问题

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..

In a consumer society, many adults channel creativity into buying things

Economic stress and unpredictable times have resulted in a booming industry for self-help products

People born without creativity never can develop it

A product has a selling price of $20, a contribution margin ratio of 40% and fixed cost of $120,000. To make a profit of $30,000. The number of units that must be sold is: Type the number without $ and a comma. Eg: 20000

Which of the following statement regarding cost is correct:

更多留学生实用工具

考试浏览器助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单