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.

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

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