Question

If the marginal probability density function of the kth order statistic of a random sample of size 8 from a uniform distribution on [0, 2] is

\[ f(x) = \begin{cases} \frac{7}{32} x^6 (2 - x), & 0<x<2 \\ 0, & \text{otherwise} \end{cases} \]

then \( k \) equals _________ (round off to 2 decimal places).

Hint:

For order statistics, the density function often takes a specific form depending on the sample size and the order \( k \).

Answer 1

Correct Answer: 7

Given the marginal probability density function, we can recognize that this corresponds to the distribution of an order statistic from a uniform distribution. By analyzing the form of the density function, we find that: \[ k = 7. \] Thus, the value is \( 7 \).