Question

Problem 10 Let $X$ be a random variable with an $F_{p,q}$ distribution. a. Let $\chi_1^2 \sim \chi^2(p)$ and $\chi_2^2 \sim \chi^2(q)$. Use $X = \frac{\chi_1^2/p}{\chi_2^2/q}$ to derive the PDF of $X$. b. Derive the mean and the variance of $X$ from the moments forms of $\chi_1^2$ and $\chi_2^2$.

          Problem 10
Let $X$ be a random variable with an $F_{p,q}$ distribution.
a. Let $\chi_1^2 \sim \chi^2(p)$ and $\chi_2^2 \sim \chi^2(q)$. Use $X = \frac{\chi_1^2/p}{\chi_2^2/q}$ to derive the PDF of $X$.
b. Derive the mean and the variance of $X$ from the moments forms of $\chi_1^2$ and $\chi_2^2$.

Problem 10
Let X be a random variable with an Fp,q distribution.
a. Let χ1^2 ∼χ^2(p) and χ2^2 ∼χ^2(q). Use X = (χ1^2/p)/(χ2^2/q) to derive the PDF of X.
b. Derive the mean and the variance of X from the moments forms of χ1^2 and χ2^2.

Added by Kevin F.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Sri K

02/21/2023

Step 1

$$ The PDF of $X_2^2$ is given by: $$f_{X_2^2}(y) = \frac{1}{2^{q/2}\Gamma(q/2)}y^{q/2-1}e^{-y/2}, \quad y>0.$$ Show more…

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Problem 10 Let X be a random variable with an $F_{p,q}$ distribution. a. Let $X_1^2 \sim \chi^2(p)$ and $X_2^2 \sim \chi^2(q)$. Use $X = \frac{X_1^2/p}{X_2^2/q}$ to derive the PDF of X. b. Derive the mean and the variance of X from the moments forms of $X_1^2$ and $X_2^2$.