 My Math Forum Sampling Distribution of Normal Random Variables
 User Name Remember Me? Password

 Advanced Statistics Advanced Probability and Statistics Math Forum

 January 29th, 2018, 06:16 PM #1 Senior Member   Joined: Oct 2015 From: Antarctica Posts: 128 Thanks: 0 Sampling Distribution of Normal Random Variables Let $X_1,X_2,...,X_m$ be i.i.d. from a $N(\mu_1,\sigma_1^2)$ distribution, and let $Y_1,Y_2,...,Y_n$ be i.i.d. from a $N(\mu_2,\sigma_2^2)$ distribution, and let the $X_i$'s be independent from the $Y_j$'s. Determine the sampling distribution of the following quantity: $$R=\frac{(m-1)S_X^2+(n-1)S_Y^2}{\sigma^2}$$ under the condition that $\sigma_1=\sigma_2=\sigma$, where $S_X^2$ and $S_Y^2$ are the respective sample variances of $X$ and $Y$. My intuition tells me that the sampling distribution is that of a chi squared with $m+n-2$ degrees of freedom, but I have absolutely no justification so it's probably wrong. So what is the sampling distribution of $R$, and how do you justify it? Last edited by John Travolski; January 29th, 2018 at 06:43 PM. January 30th, 2018, 11:59 AM #2 Senior Member   Joined: Sep 2015 From: USA Posts: 2,529 Thanks: 1389 what formula for sample variance are you using? If you are using the unbiased version, i.e. $S_X^2 = \dfrac{1}{K-1}\sum \limits_{k=1}^K~(X_m-\mu)^2$ Then $\dfrac{m-1}{\sigma^2}S_X^2 = (m-1)\left[\dfrac{1}{m-1}\sum \limits_{k=1}^m ~\left(\dfrac{(X_k-\mu_1)}{\sigma}\right)^2\right] = \sum \limits_{k=1}^m ~\left(\dfrac{(X_k-\mu_1)}{\sigma}\right)^2$ This is seen to be the sum of the squares of $m$ standard normals. Similarly the $Y$'s produce the sum of squares of $n$ standard normals. Thus $R$ is Chi-Square with $(m+n)$ degrees of freedom. If you use the biased version of the sample variance you get these $\dfrac{m-1}{m}$ and $\dfrac{n-1}{n}$ factors which make things a mess. January 31st, 2018, 04:05 PM #3 Senior Member   Joined: Oct 2015 From: Antarctica Posts: 128 Thanks: 0 I was using the unbiased version, that helps out a lot. Thank you. Tags distribution, normal, random, sampling, variables Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post chr Probability and Statistics 2 January 27th, 2018 10:57 PM flowe Probability and Statistics 6 July 14th, 2017 01:20 PM mia6 Advanced Statistics 1 February 23rd, 2011 01:11 PM callkalpa Advanced Statistics 0 May 17th, 2010 06:47 AM StrangeAttractor Probability and Statistics 2 August 24th, 2008 08:26 AM

 Contact - Home - Forums - Cryptocurrency Forum - Top

Copyright © 2019 My Math Forum. All rights reserved.       