Daily Normal RV

You are drawing from a normally distributed random variable (RV) $X \sim N(0, 1)$ once a day. What is the approximate expected number of days until you get a value greater than 2?

This is the same question as problem #16 in the Statistics Chapter of Ace the Data Science Interview!

Since $X$ is normally distributed, we can employ the cumulative distribution function (CDF) of the normal distribution:

$\Phi(x) = P(X \le x)$

Knowing that *X* has a standard normal probability distribution, we can use the CDF to find the probability of *X* being at least 2:

$\Phi(2) = P(X \le 2) = P(X \le \mu + 2\sigma) = 0.9772$

Therefore, $P(X > 2) = 1 - 0.977 = 0.023$ for any given day. Since each day's draws are independent, the expected time until drawing an $X > 2$ follows a geometric distribution, with $p = 0.023$. Letting $T$ be a random variable denoting the number of days, we have the following:

$E[T] = \frac{1}{p} = \frac{1}{.0228} \approx 43 \space \text{days}$