Question 1 Of 10: Which Of The Following Is True?

Question 1 Of 10100 Pointswhich Of The Following Is True Is X Is A Co

Question 1 of 10.0 Points: Which of the following is true if X is a continuous random variable, and f(x) is the probability density function of X?

A. f(x) ≥ 0

B. All of the given statements are true

C. The area under the entire graph of f(x) equals 1

D. For any two numbers a and b with a ≤ b, P(a ≤ X ≤ b) = ∫_a^b f(x) dx

Paper For Above instruction

Understanding the fundamental properties of continuous random variables and their probability density functions (pdf) is essential in statistics, particularly in probability theory and statistical inference. The question at hand evaluates the core attributes that define a valid probability density function for a continuous random variable, such as non-negativity, total integral equating to one, and the relationship between the pdf and probabilities over intervals.

Firstly, for any continuous random variable X with pdf f(x), the non-negativity condition f(x) ≥ 0 is a basic property inherent to all probability density functions. This ensures that probabilities, which are derived from the integrals of f(x), are not negative, adhering to the axioms of probability theory. Therefore, statement A, f(x) ≥ 0, is always true for a valid pdf.

Secondly, the integral of f(x) over its entire domain must be equal to 1, which translates to the total probability of the entire sample space being 1. This is a fundamental requirement for any probability distribution, ensuring that the total probability encompasses all possible outcomes. Statement C captures this property explicitly and is a necessary condition for a function to qualify as a pdf.

Thirdly, the probability that X falls within the interval [a, b] is given by integrating the pdf over that interval, expressed mathematically as P(a ≤ X ≤ b) = ∫_a^b f(x) dx. This relationship underpins the connection between the density function and probability measures across intervals. Statement D states this correctly, reaffirming the definition of the probability density function.

Finally, while statement B claims that all the given statements are true, considering the individual statements demonstrates that each one correctly describes properties of a pdf. As such, this statement is also accurate, provided the context is the set of all true properties of a pdf.

In conclusion, all options detailed in the question are valid properties of a probability density function for a continuous random variable. Specifically, the non-negativity of f(x), the total integral being one, and the relationship between the pdf and probabilities over intervals are fundamental features that validate a function as a pdf. This understanding is critical for accurately modeling continuous random phenomena and calculating associated probabilities.

Answer

The correct answers are B, C, and D. These encompass the fundamental properties that define a valid probability density function for a continuous random variable. Specifically, f(x) ≥ 0 (A), the total area under the curve equals 1 (C), and the probability over an interval is represented by the integral of f(x) over that interval (D). As all these statements are true and correctly describe features of a pdf, the most comprehensive choice—"All of the given statements are true"—is also correct (B).

References

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