Division of Chi Squared Random Variables

Click For Summary
SUMMARY

The division of two independent chi-squared random variables, X and Y, each with n degrees of freedom, does not yield a standard distribution of 1. Instead, the ratio X/Y follows an F-distribution, specifically derived from the properties of chi-squared distributions. It is crucial to understand that having the same degrees of freedom does not imply that X and Y are equal or follow the same distribution. For a comprehensive understanding, one should refer to the derivation of the F-distribution.

PREREQUISITES
  • Understanding of chi-squared distributions
  • Knowledge of F-distribution properties
  • Familiarity with statistical independence
  • Basic concepts of probability density functions (PDFs)
NEXT STEPS
  • Study the derivation of the F-distribution from chi-squared distributions
  • Explore the properties of independent random variables in statistics
  • Learn about the applications of F-distribution in hypothesis testing
  • Investigate the implications of variable independence in statistical analysis
USEFUL FOR

Statisticians, data analysts, and students studying probability theory who seek to deepen their understanding of chi-squared and F-distributions.

jojay99
Messages
10
Reaction score
0
Hey guys,

I have a quick question. Suppose X is a chi squared random variable with n degrees of freedom and Y is another independent chi squared random variable with n degrees of freedom.

Is X/Y ~ 1 ?

Intuitively, it makes sense to me but I'm not too sure.
 
Physics news on Phys.org


Hey jojay99 and welcome to the forums.

If X != Y the answer is an emphatic no. You can't just assume that X = Y just because both have the same degree of freedom: if they are two distinct random variables then they will have some distribution.

The result you should look at is the derivation of the F-distribution:

http://en.wikipedia.org/wiki/F-distribution

Any derivation of the F-distribution will tell you how the distribution for a ratio of chi-square distributions (with terms for the degrees which are constants) is derived.

Remember that if you have two distributions you need to check whether the two variables correspond to the same process and not the same variable definition or PDF.

Another thing to think about: is X + Y = 2X? How about X + Y = 2Y? Even if it's not a random variable, just think a normal variable and consider those questions.
 

Similar threads

Undergrad Randomly Stopped Sums vs the sum of I.I.D. Random Variables
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Karhunen–Loève theorem expansion random variables
  • · Replies 1 ·
Replies
1
Views
2K
High School Chi-Square Tests for Homogeneity & Association
  • · Replies 7 ·
Replies
7
Views
3K
Graduate Sum of independent random variables and Normalization
  • · Replies 10 ·
Replies
10
Views
3K
MHB Distribution and Density functions of maximum of random variables
  • · Replies 6 ·
Replies
6
Views
5K
MHB Derivation of snedecors F distribution
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad Distribution of ratio std Normal and sqrt chi squared
  • · Replies 4 ·
Replies
4
Views
4K
High School How do we interpret a random variable?
  • · Replies 10 ·
Replies
10
Views
2K
Undergrad Is this conditional expectation identity true?
  • · Replies 1 ·
Replies
1
Views
2K
Graduate Difficulty with summation of non-central chi-squared random variables
  • · Replies 3 ·
Replies
3
Views
3K