# Let X \sim N(6,4).Find the probabilities P(5<X<7).

Let $X\sim N\left(6,4\right)$.Find the probabilities $P\left(5.

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casincal

Step 1
Introduction:
The normal probability is a type of continuous probability distribution that can take random values. The normal distribution is determined by the two parameters - the population mean $\left(\mu \right)$ and population variance $\left({\sigma }^{2}\right)$. It is symmetric with respect to its mean.
Given information:
$X\sim N\left(6,4\right)$
Therefore,
$\mu =6$
${\sigma }^{2}=4$
Step 2
$P\left(5 is computed as follows:
$P\left(5
$=P\left(\frac{X-\mu }{\sqrt{{\sigma }^{2}}}<\frac{7-\mu }{\sqrt{{\sigma }^{2}}}\right)-P\left(\frac{X-\mu }{\sqrt{{\sigma }^{2}}}<\frac{5-\mu }{\sqrt{{\sigma }^{2}}}\right)$
$=P\left(Z<\frac{7-6}{\sqrt{4}}\right)-P\left(Z<\frac{5-6}{\sqrt{4}}\right)$
$=P\left(Z<0.5\right)-P\left(Z<-0.5\right)$
$=P\left(Z<0.5\right)-\left[1-P\left(Z<0.5\right)\right]$
$=0.69146-\left(1-0.69146\right)$
$=0.69146-0.30854=0.38292$
Therefore,
$P\left(5