# The random variable X follows a normal distribution ?(20, 102).Find P(10 < ? < 35),

The random variable X follows a normal distribution $?\left(20,102\right)$.
Find $P\left(10,

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Step 1
Normal probability is a type of continuous probability distribution that can take random values on the whole real line. The main properties of the normal distribution are:
-It is continuous (and as a consequence, the probability of getting any single, specific outcome is zero)
-It has a "bell shaped" distribution (and that is where the "Bell-Curve" name comes along)
-The normal distribution is determined by two parameters: the population mean and population standard deviation
-It is symmetric with respect to its mean.
Given : The random variable X follows a normal distribution .
Notation: $X\sim N\left(\mu =20,{\sigma }^{2}=102\right)$
Step 2
We need to compute $Pr\left(10\le X\le 35\right)$.
The corresponding z-values needed to be computed are:
${Z}_{1}=\frac{{X}_{1}-\mu }{\sigma }=\frac{10-20}{10.1}=-0.9901$
${Z}_{2}=\frac{{X}_{2}-\mu }{\sigma }=\frac{35-20}{10.1}=1.4851$
Therefore, we get:
$Pr\left(10\le X\le 35\right)=Pr\left(\frac{10-20}{10.1}\le Z\le \frac{35-20}{10.1}\right)=Pr\left(-0.9901\le Z\le 1.4851\right)$
$=Pr\left(Z\le 1.4851\right)-Pr\left(Z\le -0.9901\right)=0.9312-0.1611=0.7702$
$Pr\left(10\le X\le 35\right)=0.7702$
Jeffrey Jordon