Normal Distribution

In probability theory, the normal (or Gaussian or Gauss or Laplace–Gauss) distribution is a very common continuous probability distribution. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. A random variable with a Gaussian distribution is said to be normally distributed and is called a normal deviate

The normal distribution is sometimes informally called the bell curve. However, many other distributions are bell-shaped (such as the Cauchy, Student’s t, and logistic distributions).

Normal distribution function

When random variable X has normal distribution,

X ∼ N (μ,σ2)

E (X) = μX

Var (X) = σ2X

The probability density function and cumulative distribution function of the normal distribution:

Probability density function (pdf)

The probability density function is given by:

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X is the random variable.

μ is the mean value.

σ is the standard deviation (std) value.

e = 2.7182818… constant.

π = 3.1415926… constant.

Cumulative distribution function

The cumulative distribution function is given by:

X is the random variable.

μ is the mean value.

σ is the standard deviation (std) value.

e = 2.7182818… constant.

π = 3.1415926… constant.

Standard normal distribution function

When

E(Z) = μz = 0

Var (Z) = σ2X = 1

Then the probability density function and cumulative distribution function of the standard normal distribution:
Probability density function

Cumulative distribution function

\Phi _{Z}(z) = P(Z\leq z)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{x}e^{\frac{-y^{2}}{2}}dy

Standard normal distribution table

Standard Normal Distribution Graph

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