Proof: Squared Standard Normal is Chi-Squared (Using CDF)

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Proof that the square of a standard normal random variable is chi-squared random variable with 1 degree of freedom. This proof uses the method of distribution functions.

See PDF of all proofs here:
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In 2:53, why it's d/dA not d/da?

ZLYang
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What's the distribution of Y=u^2 +v^2 given that u and v follow a standard normal distribution

frankmusonda
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A. Thank you!
B. Maybe it is due to my screen setting, but the pen color seems faint.

kottelkannim