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 random variable that assigns equal likelihood to every outcome in an interval is the continuous uniform distribution.

1. Relevant concepts
- Continuous random variable: described by a probability density function (PDF) f(x) such that P(X in [c,d]) = ∫_c^d f(x) dx.
- Equal likelihood on [a,b] means the PDF is constant on that interval and zero outside.
- Continuous uniform distribution on [a,b] has PDF f(x) = 1/(b−a) for a ≤ x ≤ b, and 0 otherwise.

2. Step-by-step reasoning
Equal likelihood across all points in a finite interval implies the PDF must be constant over that interval. The only standard continuous distribution with a constant nonzero PDF on a finite interval is the uniform distribution.

3. Calculations and intermediate steps
Normalization: ∫_a^b f(x) dx = 1, with constant f(x) = c, gives c(b − a) = 1 ⇒ c = 1/(b − a).  
Mean: μ = E[X] = ∫_a^b x·(1/(b−a)) dx = (1/(b−a))·[x^2/2]_a^b = (b^2 − a^2)/(2(b−a)) = (a + b)/2.  
Second moment: E[X^2] = (1/(b−a))·[x^3/3]_a^b = (b^3 − a^3)/(3(b−a)) = (a^2 + ab + b^2)/3.  
Variance: Var(X) = E[X^2] − μ^2 = (a^2 + ab + b^2)/3 − (a + b)^2/4 = (b − a)^2/12.

4. Why the correct answer is correct
A constant PDF on [a,b] directly embodies “all outcomes between some minimum and maximum are equally likely.” The uniform distribution is defined exactly by this property.

5. Why the other choices are incorrect
- Lognormal: skewed, positive-only, not constant density.  
- Normal: bell-shaped, probabilities vary across the interval (not equal).  
- Exponential: decaying PDF (memoryless), defined on [0, ∞), not constant.

6. Conclusion
Therefore, the distribution described is the uniform distribution, matching the provided correct 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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