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Normal Distribution:
Normal Distribution for a random variable X with mean and standard deviation is given by
f(x)=12e–12(x-)2
A continuous random variable X is said to follow a normal distribution with parameter and if the probability function is f(x)=12e–12(x-)2; –<x<, –<<, and >0.
XN(,) denotes that the random variable X follows a normal distribution with a mean and standard deviation.
We can even write the normal distribution as XN(,) denoting the normal distribution as XN(,2) symbolically.
Constants of Normal Distribution:
Mean =μ
Variance = ó²
Standard deviation = ó
A normal curve graph is shown above.
The Properties of Normal Distribution:
- The normal curve is bell-shaped.
- It is symmetrical about the line, X= ie., about the mean line.
- Mean= Median=Mode=
- The height of the normal curve is maximum at X= and 12 is the maximum height.
- It has only one mode at X=. Since the normal curve is unimodal
- The points of inflection are at X=
- Since the curve is symmetrical, about X=, skewness is zero.
Solved examples:
- Find the probability function for a normal distribution where the mean is 4 and the standard deviation is 2 and x = 3.
Given:
Mean =4
Standard Deviation =2
X=3
The normal distribution function is given by
f(x)=12e–12(x-)2
f(x)=122e–12(3-42)2
f(x)=0.19947e-0.125
f(x)=0.199470.882496
f(x)=0.17603
Hence, the probability density function for normal distribution is 0.17603.
