// Numbas version: finer_feedback_settings {"name": "Chi-square and Normal transformation", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Chi-square and Normal transformation", "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "

Let $X \\sim N(0, 1)$ (standard Normal distribution)

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And $Y = X^2$

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Then $Y \\sim \\chi^2(1)$ (Chi-squared distribution with 1 degree of freedom)

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If $Z = \\sqrt{Y}$ why is $Z$ not the standard normal distribution?

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Taking the square of a standard normal produces a chi-square random variable.

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But taking the square root of the chi-square won't produce a normal since the range of the resulting distribution will be $(0, \\infty)$ not the entire set of real numbers and the PDF of $Z$ is $2$ times the PDF of $X$ so it still integrates to $1$.

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Performing both of these transformations is a good exercise.

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