Probability Distribution Function of Discrete and Continues Random Variable

Probability distribution function.

A probability distribution function is a probability discrete measure of all sort

example1 

Find the constant alpha(α) in the function below.

Fₓ(x) = {₀ᵅˣ  x = 1, 2, 3

the function above illustrate probability distribution function of discrete random variable.

but ∑xf(x)=1

The above expression is used whenever a constant is attached in discrete probability distribution function. Now we're all looking for alpha α.

α = α(1)+α(2)+α(3)=1

α = α6 but α6=1

α= ⅙

Our new function is  Fₓ(x) = {¹/6 (x)  but x = 1 2 3

Since we all know that probability must be equal to one. let's see if our solution would be equal to one.

(1/6)x + (1/6)x + (1/6)x = 1

(1/6)1 + (1/6)2 + (1/6)3 = 6/6 = 1

Introduction to probability complete course.

Hence proved its probability.

Probability distribution function of continues random variable.

The probability distribution function of continues random variable.

In solving the continues case, the summation ∑ wil be change to integral ∫ therefore the equation for finding the constant is ∫ (fxdx)=1

Example

Suppose we are given ∫ cx at a range of 0≤x≤1 where a = 0 and b = 1 by applying the a and b upon the integral symbol we have ∫ₐᵇ, now let's solve to find the constant C.

∫cxdx=1 0≤x≤1 

∫ₐᵇ cxdx =1

∫¹₀ cx²/2 =1

by integration  C [¹/2 - ⁰/2] = 1

C/2 =1

therefore C =2

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