Normal Distribution question. Finding mean

Question

The random variable $x$ has a normal distribution. The mean is $\mu$ ( where $\mu \gt 0$ ) and the variance is $\frac{1}{4}\mu ^2$

Find $P( x> 1.5\mu )$

Please help me with this question! Been struggling with it for some time .

Answer 1

\begin{equation}Pr(X>1.5\mu)\\=1-Pr (X <1.5\mu)\\=1-\phi (\frac{1.5\mu -\mu}{\sqrt {.25}\mu})\end{equation} where $\phi $ is the standard normal CDF. Simplify this (for non-zero $\mu $ and use statistical tables).

Answer 2

We know $\sigma = \frac{\mu}{2}$. So,

$$P(X> 1.5 \mu) = P(X-\mu > \frac{\mu}{2}) = P(X-\mu > \sigma) = P(\frac{X-\mu}{\sigma} >1) = P(Z>1)$$ where $Z\sim N(0,1)$