Suppose that X is a normal random variable with mean 5. If P{X>9}=.2, approximat

Suppose that X is a normal random variable with mean 5. If P{X>9}=.2, approximately what is Var(X)?
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Ched1950

Given: X is a normal random variable with mean $\mu =5$ and $P\left[X>9\right]=0.2$
To find: variance of X
Solution: Let ${\sigma }^{2}$ be the variance of X. Now,
$P\left[X>9\right]=0.2$
$⇒P\left[X\le 9\right]=1-0.2=0.8$
$⇒P\left[\frac{x-\mu }{\sigma }\le \frac{9-4}{\sigma }\right]=0.8$
$⇒P\left[Z\le \frac{4}{\sigma }\right]=0.8$
$⇒\varphi \left(\frac{4}{\sigma }\right)=0.8$
$⇒\frac{4}{\sigma }=0.845$
$⇒\sigma =\frac{4}{0.845}=4.73$
Thus variance of X is $var\left(X\right)={\sigma }^{2}={\left(4.73\right)}^{2}=22.37=22$