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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-

$1)$ $P(-3<X<0)$
$2)$ $F^{-1}(1/4)$

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 $2)$

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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 $z$

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