Question

Use definition 3.5 and the result in part (b) to show that v(x) = e{[(x − e(x)]²} = e[(y − μ)²] = σ²; that is, x = y 1 and y have equal variances.

a) Probability theory
b) Linear algebra
c) Differential equations
d) Statistics

Answer 1

This question is a College-level statistics problem. It requires using the mean and variance formulas for uniform and exponential distributions to show that the variables x and y have equal variances, specified in terms of standard deviation and mean.

Step-by-step explanation:

The subject of this question appears to be from the field of statistics, specifically focusing on probability distributions, expected value, and variance. The question involves demonstrating that random variables x and y have equal variances using a theoretical distribution of X ∼ U(0,1) and given statistical properties such as mean (μ) and variance (σ²).

The mean (μ) of a uniform distribution U(a, b) is calculated using the formula μ = (a + b) / 2. The variance (σ²) for a uniform distribution U(a, b) is calculated using the formula σ² = ((b - a)² / 12).

Since x and y are said to have equal variances, and given that x is a uniform distribution, y will also follow the same pattern, and thus will have the same standard deviation as the mean, eligible under certain conditions of the exponential distribution as mentioned, where X ∼ Exp(0.25) and the probability density function is f(x) = me-mx.

The question seems to require the application of these concepts to demonstrate the relationship between the variance of the variable v(x), which would involve equating to σ², and verifying that this variance equals that of y given that both have the same distribution parameters.