2X Exceeding Y

Solution & Explanation

To solve the problem of finding the probability that $2X > Y$ where $X$ and $Y$ are independent standard normal random variables, we need to consider the transformation of these variables and the properties of normal distributions.

Step-by-Step Explanation

1. Understanding the Variables:

   • $X$ and $Y$ are standard normal random variables. This means $X \sim N(0, 1)$ and $Y \sim N(0, 1)$.
   • We are tasked with finding $P(2X > Y)$.

2. Reformulating the Problem:

   • The inequality $2X > Y$ can be rewritten as $2X - Y > 0$.
   • We need to find $P(2X - Y > 0)$.

3. Distribution of the Linear Combination:

   • Since $X$ and $Y$ are independent, the linear combination $2X - Y$ will also be normally distributed.
   • The mean of $2X - Y$ is given by: $E(2X - Y) = 2E(X) - E(Y) = 2 \times 0 - 0 = 0$
   • The variance of $2X - Y$ is given by: $\text{Var}(2X - Y) = 4 \times \text{Var}(X) + \text{Var}(Y) = 4 \times 1 + 1 = 5$
   • Therefore, $2X - Y \sim N(0, 5)$.

4. Symmetry of the Normal Distribution:

   • The normal distribution $N(0, 5)$ is symmetric about the mean, which is 0.
   • The probability that a normally distributed variable is greater than its mean is 0.5 due to this symmetry.

5. Conclusion:

   • Thus, the probability that $2X > Y$ is: $P(2X - Y > 0) = 0.5$

This result stems from the properties of the normal distribution and the transformation of the original variables into a new normally distributed variable. The symmetry of the normal distribution around its mean ensures that the probability of being greater than the mean is equal to the probability of being less than the mean, each being 0.5.