Which of the following characterizes a continuous random variable for which all outcomes between some minimum and maximum value are equally likely?

Question

BlackTom AI · Accepted Answer

A continuous uniform distribution assigns a constant probability density to every point in the interval [a, b], so every outcome between the minimum a and maximum b is equally likely.

1. Relevant knowledge points and concepts
- Continuous random variable: described by a probability density function (pdf) f(x) such that probabilities are integrals of f(x).
- Uniform distribution on [a, b]: pdf is constant on [a, b] and zero outside. Key properties: constant density, finite support, equal likelihood for all values in the interval.
- Contrasting distributions: exponential is one-sided and decays; lognormal is positively skewed; normal is unbounded and bell-shaped.

2. Complete reasoning process step by step
- "Equally likely between some minimum and maximum" implies the pdf must be constant on that interval and zero elsewhere.
- Let the interval be [a, b]. A constant pdf f(x) = c for x in [a,b] must satisfy the normalization ∫_a^b c dx = 1, which yields c(b − a) = 1, so c = 1/(b − a).
- Therefore the pdf is f(x) = 1/(b − a) for a ≤ x ≤ b and f(x) = 0 otherwise.

3. Calculations and intermediate steps
- Normalize: ∫_a^b f(x) dx = ∫_a^b c dx = c(b − a) = 1 ⇒ c = 1/(b − a).
- Mean: μ = ∫_a^b x f(x) dx = ∫_a^b x/(b − a) dx = [x^2/(2(b − a))]_a^b = (a + b)/2.
- Variance: σ^2 = (b − a)^2/12 (can be derived from moments).

4. Why the correct answer is correct
- The requirement "all outcomes between some minimum and maximum are equally likely" matches exactly the definition of the continuous uniform distribution: constant density over [a,b], equal probability per unit length.

5. Why other options are incorrect
- Exponential distribution: support [0, ∞) with decreasing density λe^{−λx}, not constant or bounded.
- Lognormal distribution: support (0, ∞) and skewed; values are not equally likely across a finite interval.
- Normal distribution: unbounded support (−∞, ∞) with a bell-shaped (nonconstant) density; values near the mean are more likely.

6. Conclusion
The correct choice is uniform distribution, which matches the provided answer: uniform distribution.

Which of the following characterizes a continuous random variable for which all outcomes between some minimum and maximum value are equally likely?单项选择题

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