Probability Distribution Transformation

The Probability distribution transformation (PDT) is a occupation of continuous random variable .probability distribution transformation is worn to get the probability of the random variable which takes a rate in a given interval. The point of the Normal Distribution arc can be find out by this Probability distribution transformation. The normal distribution of the Continuous Probability distribution transformation is called the Gaussian Function.A normal distribution which have a mean=0 and a standard deviation=σ in a arbitrary variable is called a standard deviation.

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Formulas for Probability Distribution Transformation:

Formula to find probability distribution transformation:

Probability distribution transformation in normal distribution= `(1/(sigmasqrt(2pi)))e-(x-m)^2 / (2sigma^2)`

Probability distribution transformation in standard normal distribution= `(1/sqrt(2pi))e-(x^2 / 2)`

Where m denotes the mean value

Where σ denotes the standard deviation value

Where π denotes the value of 3.14

Where e denotes the value of 2.718


Example for Probability Distribution Transformation:

Example 1:

Find the probability distribution transformation where

mean m=2
Standard deviation σ=1

Normal random variable x=4

Solution:

Step 1: To find Probability density function first find sqrt(2π).
`sqrt(2pi)` = `sqrt(2 xx 3.14)`
= `sqrt(6.28) ` = 2.51

Step 2: To Find `1/(sigmasqrt(2pi)).`
` sigmasqrt(2pi)` = 1 x 2.51 = 2.51
`1/(sigmasqrt(2pi))` = 1/2.51 = 0.398

Step 3: To Find `e-(x-m)^2 / (2sigma^2)` calculate` -(x-m)^2` and `2sigma^2.`
`-(x-m)^2 ` = `-(4-2)^2`
= 22 = 4
` 2sigma^2` = 2 x (12)
= 2 x 1 = 2
`-(x-m)^2 / (2sigma^2)` = 4/2
= 2

Step 4: to find `e-((x-m)^2 / (2sigma^2))`
= 7.389


Step 5:to  calculate probability density function we use the formula and we get the final solution
= 0.398 x 7.389 = 2.94

I am planning to write more post on Unit Conversion Chart and What is Associative Property. Keep checking my blog.

Example 2:

Find the Probability distribution transformation where

mean m=6
Standard deviation σ=2

Normal random variable x=10

Solution:

Step 1: To find Probability density function first find sqrt(2pi).


`sqrt(2pi) = sqrt(2 xx 3.14)`


` = sqrt(6.28) = 2.51`

Step 2: To Find  `1/(sigmasqrt(2pi))`


` sigmasqrt(2pi) = 2xx 2.51 = 5.02`


`1/(sigmasqrt(2pi)) = 1/5.02 = 0.199`

Step 3: To Find` e-((x-m)^2 / (2sigma^2))` calculate `-(x-m)^2` and `2sigma^2`

` -(x-m)^2 ` = 4^2 = 16
` 2sigma^2` = 2 x (22)
= 2 x 4= 8
`-(x-m)^2 / (2sigma^2)` = 16/8
= 2

Step 4: to find `e-((x-m)^2 / (2sigma^2))`
= 2.7182 = 7.387


Step 5:to  calculate probability density function we use the formula and we get the final solution
= 0.199 x 7.3887= 35.810