GATE CS 2008 Q29 — Probability: Random Variables and Distributions

We are given two normally distributed random variables:

• $X \sim N(\mu_X = 1, \sigma_X^2 = 4)$, which gives the standard deviation $\sigma_X = 2$.
• $Y \sim N(\mu_Y = -1, \sigma_Y^2)$, where the standard deviation $\sigma_Y$ is unknown.

We are given the probability equality:

To solve this, we standardize both variables to the standard normal variable $Z \sim N(0, 1)$ using the transformation $Z = \frac{W - \mu}{\sigma}$:

1. Standardizing $X$: $$P(X \leq -1) = P\left( \frac{X - \mu_X}{\sigma_X} \leq \frac{-1 - 1}{2} \right) = P(Z \leq -1)$$
2. Standardizing $Y$: $$P(Y \geq 2) = P\left( \frac{Y - \mu_Y}{\sigma_Y} \geq \frac{2 - (-1)}{\sigma_Y} \right) = P\left( Z \geq \frac{3}{\sigma_Y} \right)$$

Equating the two standardized probabilities:

By the symmetry of the standard normal distribution, we know that $P(Z \leq -z) = P(Z \geq z)$ for any real number $z$. Therefore, we can directly equate the limits:

Thus, the standard deviation of $Y$ is 3.