Question

Compute \(P\left(Y_{3}<\xi_{0.5}

Short answer

\(\frac{4}{9}\)

Step-by-step solution

Define Variables

Let \(Y_1, Y_2, ..., Y_9\) represent the order statistics of the sample of size 9. \(\xi_{0.5}\) represents the median of the continuous distribution.

Recall Property of Order Statistics

For a random sample of size \(n\) from a continuous distribution, any particular order statistic has the distribution of a Uniform[0,1] random variable. Essentially, there's an equal chance for any sample value to be in any position within the ordered set.

Determine Probability

From our knowledge of uniform distribution, we know that the probability a variable takes a value between \(a\) and \(b\) on the interval [0,1] is simply \(b-a\). Therefore, the probability of the median (also a uniformly distributed variable) falling between \(Y_3\) and \(Y_7\) is essentially asking for the probability of a Uniform[0,1] variable \(U\) falling between 3/9 and 7/9, which is \(P\left(\frac{3}{9} < U < \frac{7}{9}\right)\). This calculates to \(\frac{7}{9} - \frac{3}{9} = \frac{4}{9}\).