# Calculating probabilities in normal distribution

If $$X$$ is a normal random variable with $$\mu=-2$$ and $$\sigma=3$$ , and has probability density function $$f_x$$ and cumulative density function $$F_x$$ , calculate-

$$1)$$ $$P(-3
$$2)$$ $$F^{-1}(1/4)$$

I tried the first part by converting $$X$$ to standard normal variable $$Z= \frac{X-(-2)}{3}$$ and using the values from standard normal table , my answer was $$0.37467$$ which is incorrect.
Furthermore I have no clue for $$2)$$

## 1 Answer

Hints:

• How did you get $$0.37467$$? You should have been looking up $$\frac23 \approx 0.66667$$ and $$-\frac13 \approx -0.33333$$ in your tables. If you looked for $$0.66$$ and $$-0.33$$ then you may need more precision

• For the second part, you need to reverse the process, so you are trying to find the value in your tables giving a probability of $$\frac14=0.25$$, i.e. the $$z$$ such that $$P(Z \le z)=0.25$$. It will not be far from $$-\frac23$$. Then you need to reverse $$z= \frac{x-(-2)}{3}$$ to get a corresponding value for $$x$$ from that $$z$$