# If X is normal, P(X<0) = 0.5, and P(X>= 1) = 0.05, find P(|X|<= 0.25)

This is a normal distribution.

The example suggests that the mean is 0, because the probability of X<0 is 0.5, which when referencing the z table gives a z-score of 0.

It then suggests the z-score of 1 is 1.645, based on the same reasoning.

However, for |X|<=0.25, the answer gives a probability of 0.3182, which I can't reason out as to why.

The standard deviation derived from the z-score of 1 is just 1.645 = 1/σ, which yields σ=1/1.645

Using this standard deviation, I can calculate the z-score of 0.25 with the same z-score formula.

Z = (0.25 - 0) / (1/1.645)

This gets a z-score of 0.41125, which when checked against the z-table, yields approximately 0.6591.

Where did I go wrong in my analysis?

Hint: $P(|X|\leq 0.25) = P(X \leq 0.25) - P(X \leq -0.25)$