Normal Distribution Definition
The most important continuous probability function in statistics Normal Distribution or Gaussian Distribution, which was innovated by Gauss to find the errors in immense observations. By which the Bell curved shape is called as Gaussian function or Bell curve is used to map the statistical distribution of probability density function. Normal Distribution Formula
f(x) = $\frac{1}{\sqrt{2 \pi \sigma^{2}}}$ $e^{\frac{(x-\mu)^2}{2\sigma^{2}}}$
where,
$\mu$ = Mean,
$\sigma$ = Standard Distribution.
If mean($\mu$) = 0 and standard deviation($\sigma$) = 1, then this distribution is known to be normal distribution.
x = Normal random variable.
Observe the value of given mean, standard deviation and limits.
Step 2 :
Find the z-value for the respective random variable by using the formula
z = $\frac{x - \mu}{\sigma}$
Where,
x = Normal random variable
$\mu$ = mean
$\sigma$ = standard deviation.
Step 3 :
Now, find the area under the standard normal curve for the z-value obtained from step2 by using Z-table.
Step 4 :
Subtract area of upper limit from area of lower limit gives the normal probabilty density function.
If mean($\mu$) = 40 and standard deviation($\sigma$) = 5, then find the P(40 $\leq$ x $\leq$ 45), where x is the normal random variable.
Step 1 :Given mean($\mu$) = 40 and standard deviation($\sigma$) = 5
P(40 $\leq$ x $\leq$ 45) = ?
Step 2 :For x = 40, z = $\frac{x - \mu}{\sigma}$
z = $\frac{40 - 40}{5}$ = 0
For x = 45, z = $\frac{x - \mu}{\sigma}$
z = $\frac{45 - 40}{5}$ = 1
Step 3 :P(40 $\leq$ x $\leq$ 45) = P(0 $\leq$ x $\leq$ 1) = [area to the left of z = 1] - [area to the left of z = 0]
Step 4 :P(0 $\leq$ x $\leq$ 1) = [area to the left of z = 1] - [area to the left of z = 0]
$\Rightarrow$ 0.8413 - 0.5
$\Rightarrow$ 0.3413
Answer :P(40 $\leq$ x $\leq$ 45) = 0.3413
If mean($\mu$) = 20 and standard deviation($\sigma$) = 4, then find the P(20 $\leq$ x $\leq$ 25), where x is the normal random variable.
Step 1 :Given mean($\mu$) = 20 and standard deviation($\sigma$) = 4
P(20 $\leq$ x $\leq$ 25) = ?
Step 2 :For x = 20, z = $\frac{x - \mu}{\sigma}$
z = $\frac{20 - 20}{4}$ = 0
For x = 25, z = $\frac{x - \mu}{\sigma}$
z = $\frac{25 - 20}{4}$ = 1.25
Step 3 :P(20 $\leq$ x $\leq$ 25) = P(0 $\leq$ x $\leq$ 1.25) = [area to the left of z = 1.25] - [area to the left of z = 0]
Step 4 :P(0 $\leq$ x $\leq$ 1.25) = [area to the left of z = 1.25] - [area to the left of z = 0]
$\Rightarrow$ 0.8944 - 0.5
$\Rightarrow$ 0.3944
Answer :P(20 $\leq$ x $\leq$ 25) = 0.3944
