Final answer:
Among the given options for the properties of the uniform distribution, option D is not true; the height of the uniform distribution for any x such that a ≤ x ≤ b is not b - a, but rather 1/(b - a).
Step-by-step explanation:
The question concerns the properties of the uniform distribution, which is a type of continuous probability distribution commonly known in mathematics.
For the uniform distribution U(a, b), where a is the minimum value and b is the maximum value that the random variable X can take on:
- The height of the distribution (also known as the probability density function value) for any x outside the interval [a, b] is 0, which corresponds to options A and B.
- For any x such that a ≤ x ≤ b, the height of the distribution is a constant, not 1 (option C) or b - a (option D).
The correct height construct for a uniform distribution is calculated as 1/(b - a). Therefore, the not true statement among the options provided is option D - The height of the distribution for any x such that a ≤ x ≤ b is b - a.