Question

Let X be a standard normal random variable. Find P(X < 0). Express your answer as a decimal.

Answer 1

A standard normal random variable \( X \) has a mean of 0 and a standard deviation of 1. To find \( P(X < 0) \), we can utilize the properties of the standard normal distribution.

The standard normal distribution is symmetric around the mean (0). Thus, the probability that \( X \) is less than 0 is equal to the probability that \( X \) is greater than 0.

Since the total area under the standard normal distribution curve is 1, we have:

\[
P(X < 0) + P(X > 0) = 1
\]

Because of the symmetry:

\[
P(X < 0) = P(X > 0)
\]

Let \( P(X < 0) = p \). Therefore, we have:

\[
p + p = 1 \quad \Rightarrow \quad 2p = 1 \quad \Rightarrow \quad p = \frac{1}{2}
\]

So,

\[
P(X < 0) = 0.5
\]

Thus, the probability \( P(X < 0) \) is:

\[
\boxed{0.5}
\]