If a continuous random variable has the following probability density function

$f (x) = {\begin{matrix} k x^{3}, 0 \leq x \leq 1, \\ 0, o t h e r w i s e; \end{matrix}$

Then the value of k is ________.

Question

Question

Answer 1

Concept:

The probability density function is always positive, and it follows the below condition.

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

Calculation:

$f (x) = {\begin{matrix} k x^{3}, 0 \leq x \leq 1, \\ 0, o t h e r w i s e; \end{matrix}$

$\int_{0}^{1} k x^{3} d x = 1$

$\Rightarrow {[\frac{k x^{4}}{4}]}_{0}^{1} = 1$

$\Rightarrow \frac{k}{4} = 1 \Rightarrow k = 4$