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 \(\int_{-\infty}^{\infty}f(x)dx\) = 1

Explanation:

Given f(x) is a probability density function so 

\(\int_{-\infty}^{\infty}f(x)dx\) = 1.....(i)

(1): \(\int_{-\infty}^{\infty}f(x+1)dx\) 

    = \(\int_{-\infty}^{\infty}f(y)dy\) (let x + 1 = y ⇒ dx = dy) 

    = 1 (Using (i))

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

Option (1) is false 

(2): \(\int_{-\infty}^{\infty}f(2x)dx\) 

    = \(\frac12\)\(\int_{-\infty}^{\infty}f(y)dy\) (let 2x= y ⇒ dx = \(\frac12\)dy) 

    = \(\frac12\) (Using (i))

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

Option (2) is correct

(3): \(\int_{-\infty}^{\infty}2f(2x-1)dx\) 

    = \(\int_{-\infty}^{\infty}f(y)dy\) (let 2x - 1 = y ⇒ 2dx = dy) 

    = 1 (Using (i))

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

Option (3) is false 

(4): \(\int_{-\infty}^{\infty}3x^2f(x^3)dx\) 

    = \(\int_{-\infty}^{\infty}f(y)dy\) (let x³ = y ⇒ 3x²dx = dy) 

    = 1 (Using (i))

So 3x2f(x3) is a probability density function 

Option (4) is false