Normal Distribution

A continuous Random Variable $X$ Is said to follow normal distribution with parameters

• $\mu$ ( Mean )
• ${\sigma }^2$ ( Variance^2 ) If its probability density function is given by

$$f(x)=\frac{1}{\sigma \sqrt{2\pi }}{e}^{1/2(\frac{x-\mu }{\sigma }{)}^2}$$

For a normal Distribution

• $mean$=$median$=$mode$=$\mu$
• The probability distribution curve is symmetric about the mean ($\mu$)
• 50% of the values are less than mean
• 50% of the values are greater than mean
• Coefficient of skewness is $0$
• $Z=\frac{x-\mu }{\sigma }$ The graph of the normal distribution is given in this graph

Standard Normal Variable

A standard normal variable is a normal variable with $\mu$ = 0 and Standard Deviation $\sigma$ = 1. Its Probability Density Function is given by $\varphi (z)=\frac{1}{\sqrt{2\pi }}{e}^{z^2/2}$ The standard normal variate is obtained from

$$Z=\frac{X-\mu }{\sigma }$$

References

Information

• date: 2025.03.22
• time: 17:04