Question

1- What is the Probability Density Functions and give example?

2- what is the types of Invertible Probability Distributions such as (uniform, trinagular, exponetial....) and give example for each?

Answer 1

1.The Probability Density Functions gives the probability associated with an interval of values say (a,b)

P ( a < X < b ) .

This probability is the area under the curve of Probability Density Function ( PDF ) from a to b . So probabilities can be evaluated by integrating the PDF of the random variable . P ( a < X < b ) = F(X) dX

EXAMPLE :  P ( 0< X < 1 ), and F(X) = X²

2. Invertible Probability Distribution

Uniform

When the variable X varies uniformly between (a,b), then the reciprocal variable Y = 1 / X has the reciprocal distribution which takes values in the range (b⁻¹ ,a⁻¹) and the function is G(1/X) = (b - Y⁻¹)/(b-a).

Triangular

When the variable X varies between (a,b) lower limit a, upper limit b and mode c, where a < b and a ≤ c ≤ b. Such a distribution is called as the Triangular Probability Distribution.

Example : a=0, b=3 , c=5 , F(X) = X⁴

Exponetial

If X is an exponentially distributed random variable with rate parameter  . then Y=1/X has the following cumulative distribution function F(Y)= e^-/Y for y >0.

Example: =1, Y = X  F(Y)= e^-1/X