Question

Assume X is an independent random variable with mean, μ ,and variance, σ²

Say if the statements are TRUE or FALSE -- if FALSE, rewrite so they are correct.
(d) X³ₙ → μ³
(e) σ² =E(X²₁) −E(X₁)²
(f) σ² = 1/n ∑ⁿᵢ₌₁ X²ᵢ - X²ₙ

Answer 1

(d) FALSE; X³ₙ does not necessarily have a mean of μ³.

(e) TRUE.

(f) FALSE; the correct formula is σ² = (1/n) ∑ⁿᵢ₌₁ (Xᵢ - μ)².

Step-by-step explanation:

In statement (d), the cube of an independent random variable Xₙ does not guarantee that its mean will be the cube of the original mean (μ³). Moments of random variables do not necessarily behave linearly, and taking the cube introduces nonlinearity, making the statement false.

Statement (e) is true. The variance, σ², can be expressed as the difference between the expectation of the square of X (E(X²)) and the square of the expectation of X (E(X)²). This is a fundamental property of the variance and holds for any random variable.

In statement (f), the provided formula for the variance is incorrect. The correct formula for the variance of a sample is σ² = (1/n) ∑ⁿᵢ₌₁ (Xᵢ - μ)², where μ is the mean of the sample. The formula provided in the question incorrectly calculates the variance using the squares of individual observations without adjusting for the mean. The correct formula accounts for the squared deviations from the mean and then averages them over the sample size, reflecting the spread of the data points around the mean.