Skew Normal Distribution

The skew normal probability distribution refers the normal probability distribution. It is also called as the Gaussian distribution. In normal distribution the mean is μ and the variance is `sigma^2` . Normal distribution is the close approximation of a binomial distribution. The limiting form of Poisson distribution is said to be normal distribution probability. This article has the information about the skew normal probability distribution.

Formula used for skew normal distribution:

The formula used for plot the standard normal distribution is

Z = `(X- mu) /sigma`

Where X is the normal with mean `mu` and the variance is `sigma^2` , `sigma` is the standard deviation.

Examples for the skew normal distribution:

Example 1 for the skew normal distribution:

If X is normally distributed the mean value is 1 and its standard deviation is 6. Determine the value of P (0 ≤ X ≤ 8).

Solution:

The given mean value is 1 and the standard deviation is 6.

Z = `(X- mu)/ sigma`

When X = 0, Z = `(0- 1)/ 6`

= -`1/6`

= -0.17

When X = 8, Z = `(8- 1)/ 6`

= `7/6`

= 1.17

Therefore,

P (0 ≤ X ≤ 4) = P (-0.17 < Z < 1.17)

P (0 ≤ X ≤ 4) = P (0 < Z < 0.17) + P (0 < Z < 1.17) (due to symmetry property)

P (0 ≤ X ≤ 4) = (0.5675- 0.5) + (0.8790 - 0.5)

P (0 ≤ X ≤ 4) = 0.0675 + 0.3790

P (0 ≤ X ≤ 4) = 0.4465

The value for P (0 ≤ X ≤ 4) is 0.4465.

Example 2 for the skew normal distribution:

If X is normally distributed the mean value is 2 and its standard deviation is 4. Determine the value of P (0 ≤ X ≤ 5).

Solution:

The given mean value is 2 and the standard deviation is 4.

Z = `(X- mu)/ sigma`

When X = 0, Z = `(0- 2)/ 4`

= -`2/4`

= -0.5

When X = 5, Z = `(5- 2)/ 4`

= 3/4

= 0.75

Therefore,

P (0 ≤ X ≤ 6) = P (-0. 5 < Z < 0.75)

P (0 ≤ X ≤ 6) = P (0 < Z < -0.5) + P (0 < Z < 0.75) (due to symmetry property)

P (0 ≤ X ≤ 6) = (0.6915 - 0.5) + (0.7734- 0.5)

P (0 ≤ X ≤ 6) = 0.1915+ 0.2734

P (0 ≤ X ≤ 6) = 0.4649

The value for P (0 ≤ X ≤ 6) is 0.4649.