For the function $f(x) = a + bx$, $0 \leq x \leq 1$, to be valid probability density function, which one of the following statements is correct?

(A) $a = 1$, $b = 4$

(B) $a = 0.5$, $b = 1$

(D) $a = 1$, $b = -1$

(GATE 2017)

Answer: (B) $a = 0.5$, $b = 1$

Explanation:

For the function $f(x)$ to be a valid probability density function (PDF), the following must hold:

$\int\limits_{0}^{1} f(x) = 1$

$\Rightarrow \int\limits_{0}^{1} (a + bx) = 1$

$\Rightarrow \Big[ax + \dfrac{bx^2}{2}\Big]_{0}^{1} = 1$

$\Rightarrow a + \dfrac{b}{2} = 1$

Only option (B) $a = 0.5$, $b = 1$ satisfies the above equation.

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