A random variable is known to have a cumulative distribution function \(F_X (x)=U(x)(1-\frac{x^2}{b})\). Its density function is:

Concept:

To analyze the random variable 'x' two functions are used.

1.Probability Distribution Function [F(x)]

2.Probability Density Function [f(x)]

Cumulative Distribution Function

If f(x) is probability density function and F(X) is cumulative distribution function then relation between both of them is:

F(X) = P[X ≤ x] = \(\mathop \smallint \nolimits_{ - \infty }^x f\left( x \right)dx\) = sum of all values less than equal to x.

Probability Density Function

It indicates the distribution of the total probability of various random variables. 

\(f(x) = \frac {d F(x)}{dx}\)

Analysis:

\(f(x) = \frac{d [U(x) (1-\frac{x^2}{b})]}{dx}\)

Let U(x) is the unit step function:

\(f(x) = \frac{d U(x)}{dx}(1-\frac{x^2}{b})+U(x) \frac{d}{dx} (1-\frac{x^2}{b})\)

= \(δ(x)(1-\frac{x^2}{b})+U(x)(\frac{-2x}{b})\)

= δ(x) - 2x/b U(x)

No option seems to be correct according to the options.