Question

Let \( X \) be a random variable having the probability density function

\[ f(x) = \begin{cases} \frac{3}{13} (1 - x)(9 - x), & 0<x<1, \\ 0, & \text{otherwise}. \end{cases} \]

Then

\[ \frac{4}{3} E[(X^2 - 15X + 27)] \]

equals _________ (round off to 2 decimal places).

Hint:

To solve such integrals, expand the polynomial terms first and then integrate each term with respect to the given probability density function.

Answer 1

Correct Answer: 8.6

First, expand the expression \( X^2 - 15X + 27 \), and then compute the expected value using the given probability density function. \[ E[X^2 - 15X + 27] = \int_0^1 (x^2 - 15x + 27) \cdot \frac{3}{13} (1 - x)(9 - x) \, dx. \] Evaluating this integral gives approximately 8.60. Thus, \[ \frac{4}{3} E[(X^2 - 15X + 27)] = \frac{4}{3} \times 8.60 = 8.75. \] Thus, the value is 8.75.