Normal Distribution Inequality

Let $X$ and $Y$ be independent random variables, each following a standard normal distribution. Determine the probability $P(Y>5X)$.

Answer

Let

\[ Z = Y - 5X. \]

Since $X$ and $Y$ are independent standard normals, $Z$ is a linear combination of independent normals, hence normal. Its mean is

\[ \mathbb{E}[Z]=\mathbb{E}[Y]-5\mathbb{E}[X]=0-5\cdot 0=0. \]

Hence, by symmetry of a mean-zero normal distribution,

\[ P(Y - 5X > 0)=\frac12=0.5. \]
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