Marginal PDF for Sum of Squares: Identifying Distribution and Calculation Method

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SUMMARY

The marginal probability density function (pdf) of the sum of squares $$\sum_{i=1}^n (X_i-\overline{X})^2$$ for independent and identically distributed (i.i.d) normal variables $$X_1,\dots,X_n$$ with distribution $$N(\mu,\sigma^2)$$ follows a chi-square distribution. Specifically, this sum can be expressed in terms of standard normal variables, where $$\chi_n^2 = \sum_{i=1}^n Y_i^2$$ and $$Y_i \sim N(0,1)$$. The transformation $$X_i-\overline{X} \sim N(0,\sigma^2)$$ is crucial for establishing this relationship.

PREREQUISITES
  • Understanding of normal distributions, specifically $$N(\mu,\sigma^2)$$
  • Familiarity with the concept of marginal probability density functions
  • Knowledge of chi-square distribution and its properties
  • Ability to manipulate statistical expressions involving sums of squares
NEXT STEPS
  • Study the derivation of the chi-square distribution from normal distributions
  • Learn about the properties and applications of the chi-square distribution in statistical inference
  • Explore the concept of sample variance and its relationship to the sum of squares
  • Investigate the Central Limit Theorem and its implications for sums of random variables
USEFUL FOR

Statisticians, data analysts, and researchers in fields requiring statistical analysis, particularly those working with normal distributions and hypothesis testing.

Fermat1
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Let $$X_1,\dots,X_n$$ be i.i.d $$N(\mu,\sigma^2)$$. What is the marginal pdf of $$\sum_{i=1}^n (X_i-\overline{X})^2$$.

I'm guessing it's some sort of chi square distribution but how to find this I am unsure. Thanks
 
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Fermat said:
Let $$X_1,\dots,X_n$$ be i.i.d $$N(\mu,\sigma^2)$$. What is the marginal pdf of $$\sum_{i=1}^n (X_i-\overline{X})^2$$.

I'm guessing it's some sort of chi square distribution but how to find this I am unsure. Thanks

The chi-square distribution is defined as $$\chi_n^2 = \sum_{i=1}^n Y_i^2$$ where $Y_i \sim N(0,1)$.
We need to rewrite the expression to standard normal distributions to relate it to the chi-square distribution.
The first step has already been done, since $X_i-\overline X \sim N(0,\sigma^2)$.
 

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