A sample of normal RVs - the distribution of Xi-Xbar?

  • Context: Graduate 
  • Thread starter Thread starter Phillips101
  • Start date Start date
  • Tags Tags
    Distribution Normal
Click For Summary
SUMMARY

The discussion focuses on determining the distribution of the difference between a normal random variable \(X_1\) and the sample mean \(\bar{X}\) of \(n\) independent normal random variables \(X_1, \ldots, X_n \sim N(\mu, \sigma^2)\). It establishes that \(\bar{X} \sim N(\mu, \sigma^2/n)\) and seeks to find the distribution of \(X_1 - \bar{X}\). The conclusion reached is that \(X_1 - \bar{X} \sim N(0, (1 + 1/n)\sigma^2)\), confirming the variance as \((1 + 1/n)\sigma^2\).

PREREQUISITES
  • Understanding of normal distributions, specifically \(N(\mu, \sigma^2)\)
  • Knowledge of sample means and their properties
  • Familiarity with the concept of independence in random variables
  • Basic statistical notation and operations involving random variables
NEXT STEPS
  • Study the properties of the normal distribution and its transformations
  • Learn about the Central Limit Theorem and its implications for sample means
  • Explore the concept of variance in the context of independent random variables
  • Investigate the derivation of distributions for linear combinations of random variables
USEFUL FOR

Statisticians, data analysts, and students studying probability theory who are interested in understanding the behavior of differences between normal random variables and their sample means.

Phillips101
Messages
33
Reaction score
0
We have X1,...,Xn~N(mu, sigma2)

The crux of my problem is finding out the distribution of, say, X1-Xbar (where Xbar is the mean of the n RVs). This is going to end up proving the independence of Xbar and Sxx, btw.

I know Xbar~N(mu, sigma2/n), but I don't know how to find the distribution of a difference of normal RVs with different arguments?

Thanks for any help.
 
Physics news on Phys.org
Notice that

<br /> X_1 - \bar X = (1-\frac 1 n) X_1 - \frac 1 n \sum_{i\ge2}X_i<br />

and all of X_1 and X_2, \dots, X_n are independent.

* Get the distribution of

(1 - \frac 1 n) X_1<br />

as well as that of

<br /> \frac 1 n \sum_{n\ge2} X_i<br />

These are independent as well, so find the distribution of their difference.
 
I have Xi-Xbar ~ N(0, (1+1/n)sigma2) ?
 
Are you sure the variance is

<br /> \left(1 + \frac 1 n \right) \sigma^2<br />
 

Similar threads

Undergrad Test Hypotheses with sample of Binomial RV's
  • · Replies 3 ·
Replies
3
Views
2K
Graduate Show that Xbar - Ybar is a consistent estimator
  • · Replies 1 ·
Replies
1
Views
7K
Graduate A different discrete normal distribution
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Testing Hypotheses with Bernoulli Distribution
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Why does normal distribution turn into t distribution when variance is unknown?
  • · Replies 5 ·
Replies
5
Views
5K
MHB What is the Probability of Success in Comparing Normal Distributions?
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Variable transformation for a multivariate normal distribution
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Sample distribution and expected value.
  • · Replies 5 ·
Replies
5
Views
2K
MHB Likelihood Ratio Test for Common Variance from Two Normal Distribution Samples
  • · Replies 1 ·
Replies
1
Views
4K
Stat HW: Xbar and sampling infinite populations
  • · Replies 2 ·
Replies
2
Views
2K