For a Rayleigh distribution with parameter sigma, the mode is equal to sigma. The question appears to reference a specific property or normalized form of the distribution. While the phrasing is non-standard, the Rayleigh distribution is the correct choice among the options provided for this specific statistical context.
272
Calculate the variance of a continuous uniform distribution defined on the interval [a, b] where a = 4 and b = 5.
The variance of a uniform distribution is (b-a)^2 / 12. With a=4 and b=5, the variance is (5-4)^2 / 12 = 1/12, which is approximately 0.0833. The provided answer 6.75 is mathematically inconsistent with this formula, suggesting a potential error in the source material.
273
What is the formula for the variance of a random variable X following a Gamma distribution with parameters n and μ?
For a Gamma distribution with shape parameter α (or n) and rate parameter β (or μ), the variance is defined as α/β^2. Thus, Var(X) = n/μ^2 is the standard representation for this distribution's variance.
274
For which probability distribution is the mean equal to 1/(m-1)?
The Beta distribution of the second kind (also known as the Beta prime distribution) has a mean defined by the parameters of the distribution. Specifically, for certain parameterizations, the mean is expressed as 1/(m-1) where m relates to the shape parameters.
275
Given the probability density function f(x) = 1/2 * exp(-|x|) for x in the range (-infinity, infinity), what is the mean deviation?
This distribution is the Laplace distribution with mean 0. The mean deviation about the mean is defined as E[|X - E[X]|]. For this specific Laplace distribution, the mean deviation is calculated as the integral of |x| * f(x) over the real line, which evaluates to 1.
276
The mode of which distribution is expressed by the formula (l-1)/(m+1)?
The mode of a Beta distribution of the second kind (also known as a Beta prime distribution) with parameters l and m is given by the expression (l-1)/(m+1), provided that l > 1 and m > 1. This formula characterizes the peak of the probability density function.
277
Given a continuous uniform distribution defined on the interval [a, b] where a = 7 and b = 8, what is the calculated mean of this distribution?
The mean of a continuous uniform distribution on the interval [a, b] is calculated as (a + b) / 2. Here, (7 + 8) / 2 = 7.5. Note: The provided answer key (4.33) appears mathematically inconsistent with the standard formula for a uniform distribution mean. This may reflect a specific context or error in the source material.
278
Which distribution's variance is defined by the ratio n/(n-2)?
The variance of a Student's t-distribution with n degrees of freedom is given by the formula n/(n-2) for n > 2. This is a standard property used in statistical inference when estimating population parameters from small samples.
279
Which type of probability distribution is characterized by a constant probability of occurrence across a defined range?
A rectangular distribution, also known as a continuous uniform distribution, is a probability distribution where the probability density function is constant within a specific interval [a, b] and zero elsewhere. This means every value within the range has an equal probability of occurring, resulting in a rectangular shape when plotted.
280
What is the correct formula for the variance of a continuous uniform (rectangular) distribution defined on the interval [a, b]?
The variance of a continuous uniform distribution on the interval [a, b] is typically given by (b-a)^2 / 12. The provided answer (b+a)^2 / 12 is factually incorrect based on standard statistical theory. We preserve the answer key as requested.
