An Illustration of the Central Limit Theorem Using Chi-Square Samples
If is a random sample from a distribution with finite mean and variance , then the central limit theorem asserts that the density function of approaches a standard normal density as . If the underlying distribution is a distribution with one degree of freedom, the density function of can be derived exactly (see the details). This Demonstration compares that density (purple) with a standard normal density (blue) for various values of .
If is a random sample from a distribution with one degree of freedom and , the density function of is .