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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:

enter image description here ?

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1 Answer 1

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"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.

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  • $\begingroup$ I don't understand how a (sum of) squared standard normal variable has a pdf like the one shown above (with the gamma function). $\endgroup$
    – E. Kaufman
    Aug 2, 2020 at 3:43
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    $\begingroup$ 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 :). $\endgroup$
    – mlofton
    Aug 2, 2020 at 3:57

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