Suppose that for a certain random variable
$X$
normally distributed,
$P( 20 \le X \le 30)=0.3$

$P(X \le 30)=3 P(X \le 20)$

We can write the first equation as
$P(X \le 30)-P(X \le 20)=0.3$

The simultaneous equations can be written
jatex options:inline}P(X \le 30)-P(X \le 20)=0.3{/jatex}  (1)
$P(X \le 30)-3 P(X \le 20)=0$
(2)
(1)-(2) gives
$P(X \le 20)=0.3 \rightarrow P(X \le 20)=0.15$
then
$P(X \le 30)=3 P(X \le 20)=3 \times 0.15 = 0.45$
.
Now we have the equations
$P(X \le 20)=0.15, \; P(X \le 20)= 0.45$
.
Using normal distribution tables or a calculator gives
$\frac{20- \mu}{\sigma}=-1.036, \; \frac{30 - \mu}{\sigma}=-0.125$

Multiplying by
$\sigma$
and subtracting gives
$-10=-0.911 \sigma \rightarrow \sigma = \frac{10}{0.911}=10.97$
then
$\mu=31.37$
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