Chance one standard normal exceeds four times another

Question

$X$ and $Y$ are independent $N(0,1)$ random variables. Compute $\mathbb{P}[Y>4X]$ .

Short answer

$1/2$ . The difference $Y-4X$ is normal with mean zero, and a mean-zero normal is positive with probability exactly one half by symmetry.

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Lessons behind this question

Continuous Random Variables and Probability Density Functions
Independence
The Normal Distribution

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