If f(x) is a probability density on the real line, then which of the following is NOT a valid probability density?

Concept:

Probability density function f(x) satisfy  = 1

Explanation:

Given f(x) is a probability density function so 

 = 1.....(i)

(1):  

    =  (let x + 1 = y ⇒ dx = dy) 

    = 1 (Using (i))

So f(x + 1) is a probability density function 

Option (1) is false 

(2):  

    =  (let 2x= y ⇒ dx = dy) 

    =  (Using (i))

So f(2x) is not a valis probability density function 

Option (2) is correct

(3):  

    =  (let 2x - 1 = y ⇒ 2dx = dy) 

    = 1 (Using (i))

So 2f(2x - 1) is a probability density function 

Option (3) is false 

(4):  

    =  (let x³ = y ⇒ 3x²dx = dy) 

    = 1 (Using (i))

So 3x2f(x3) is a probability density function 

Option (4) is false