# Why Is A Squared Standard Normal Variable A Chi Square Variable

If for any $$i \in \lbrace1,2,...n\rbrace$$ where $$Z_i \sim N(0,1)$$ and all $$Z_1, Z_2, ..., Z_n$$ are independent of each other, why is it that $$Z_i^2 \sim \chi^2_1$$ and $$\sum_i Z_i^2 \sim \chi^2_n$$ when the pdf for a chi-square distribution with $$k$$ degrees of freedom is:

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## 1 Answer

"why is it..."

Because one of the first persons (Karl Pearson?) to calculate the density function of $$Z_i^2$$ and $$\sum_{i=1}^n Z_i^2$$ chose to name these random variables $$\chi^2$$ random variables with $$1$$ and $$n$$ degrees of freedom respectively. If he had chosen some other name, say $$\Phi^2$$ random variables, you would have been asking why $$Z_i^2$$ and $$\sum_{i=1}^n Z_i^2$$ are called $$\Phi^2$$ random variables.

• I don't understand how a (sum of) squared standard normal variable has a pdf like the one shown above (with the gamma function). Aug 2, 2020 at 3:43
• The proof for the case of 1 normal random variable is here. statlect.com/probability-distributions/chi-square-distribution. As the author says, the general case for $n$ normal random variables is straightforward :). Aug 2, 2020 at 3:57