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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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