Normal Distribution

 

Definition:

A random variable defined in the range - infinity to infinity is said to follow normal distribution. If its probability density function is.

In the above probability density function.

Then the probability density function of normal distribution reduces to.

Which is called the probability density function of standard normal distribution.

Properties on normal distribution:

• Normal distribution is a symmetric distribution and normal curve is bell-shaped curve.
• Since, Normal distribution is symmetric then Mean = Median = Mode.
• Since, Normal distribution is symmetric, then odd moments become zero.

• Moment generating function of normal distribution is.

• Characteristic function of normal distribution is.

• Recurrence relation of moments.

• Normal distribution satisfies the additive property.
• Mean of normal distribution is Mew and variance is Sigma Square.
• The points of inflexion of normal curve are.
  
• x-axis is an asymptote to the normal curve at 
  
• Area property of normal distribution is

• Quartile deviation: Mean deviation: Standard deviation 

• The maximum probability occurs at x=mew and the maximum probability is
  
  
  
• Binomial distribution and Poisson distribution tents to Normal distribution

Applications of Normal distribution:

1. Many discrete distributions like binomial and Poisson are approximated to normal distribution.
2. Many continuous distributions like Chi-Square, t and f tents to Normal distribution.
3. Normal distribution is in test of significance.
4. Normal distribution is used in Statistical Quality Control to find the control limits.
5. Normal distribution is used in reliability theory.

Mean and Variance of Normal distribution:

We know that the probability density function is. 

By the definition of expectation mean.

Mode of normal distribution:

We know that the probability density function of normal distribution is.

 

Mode is the solution of.

Median of normal distribution:

We know that median is the value of 'm' obtained by solving.

Moment generating function of normal distribution:

We know that the probability density function of normal distribution is.

By the definition of moment generating function,

Characteristic function of normal distribution:

We know that the probability density function of normal distribution is.

By the definition of characteristic function.

Thank you!