Number of mean and variance possible for normal distribution

Question

Let $X ∼ N(μ, σ^2)$. Find the values of μ and $σ^2$ such that $P\left(|X| < 2\right) = \frac{1}{2}$

Are the values of μ and $σ^2$ unique? Prove or disprove it.

I am able to find reach the equation $F_x(2) < \frac{3}{4}$. But I am stuck after that. Any help is highly appreciated.

Answer 1

We define the standard normal variate, $Z =\frac{X-\mu}{\sigma}$. We then have, $$P(|X|<2) = P(-2<X<2) = P(\frac{-2-\mu}{\sigma}<Z<\frac{2-\mu}{\sigma})$$ $$=P(0<Z<\frac{2-\mu}{\sigma})+P(0<Z<\frac{2+\mu}{\sigma})=\frac{1}{2}$$

Now just look at the statistical tables.