Final answer:
For an F distribution with ν2 > 2, the mean is defined and calculated as the degrees of freedom in the denominator divided by the degrees of freedom minus 2, denoted as μ = ν2 / (ν2 - 2).
Step-by-step explanation:
To show that for ν2 > 2 the mean of the F distribution is given by a certain formula, we need to consider the degrees of freedom of the distribution and its properties. The F distribution is denoted as F~ F(n₁ − 1, n₂ − 1), where n₁ and n₂ are the sample sizes of the two distributions being compared, and thus, n₁ − 1 and n₂ − 1 are the degrees of freedom for the numerator and denominator, respectively. For the F distribution, the mean is defined only when there are more than 2 degrees of freedom in the denominator (ν2 > 2), and in such cases, the mean of the F distribution is given by: μ = ν2 / (ν2 - 2). It's important to note that as the degrees of freedom increase, the F distribution becomes more symmetric and approximates a normal distribution. This is because the central limit theorem applies as the sample size grows larger, indicating that the means of the sample distributions converge towards a normal distribution.