Constructing a Chi-Square Distribution from n Uniform Distribution Samples

Constructing a Chi-Square Distribution from Uniform Distribution Samples

In statistical analysis, transforming samples from one distribution to another can simplify complex problems. One interesting transformation is deriving a chi-square distribution with degrees of freedom from uniformly distributed samples. Here's a step-by-step process of this derivation.

Step-by-Step Derivation

Step 1: Sample from Uniform Distribution

Assume we have independent random variables that are uniformly distributed on the interval :

Step 2: Transform to Exponential Distribution

We can transform these uniform variables into exponential variables using the inverse transform sampling method. For a uniform random variable , the transformation follows an exponential distribution with parameter :

Step 3: Sum of Exponential Variables

The sum of exponential random variables, each with parameter , follows a gamma distribution with shape parameter and rate parameter . Mathematically:

Step 4: Gamma to Chi-Square Distribution

We know that a gamma distribution with shape parameter and rate parameter can be transformed into a chi-square distribution if and . Hence, the gamma distribution can be expressed as a chi-square distribution with degrees of freedom:

Conclusion

By transforming uniformly distributed samples into exponential variables and summing them up, we can derive a chi-square distribution with degrees of freedom. This process illustrates the power of distribution transformations in statistical hypothesis testing and simplifies the handling of complex distributions.