Question

Let Y₁ < Y₂ < Y₃ < Y₄ < Y₅ be the order statistics of a random sample of size 5 from a continuous distribution with median m. What is P(Y₂ < m < Y₄)?

Answer 1

Let Y1 < Y2 < Y3 < Y4 < Y5 be the order statistics of a random sample of size 5 from a continuous distribution with median m.

We know that for median if the sample observation is in ascending order and if the number of sample observations n is odd then the median is th sample observation and if total number of sample observations n is even then median is the any value between th observation and th observation , for simplicity we take the average of th observation and observation.

Here the number of sample observation = n = 5.

So, observation is median. So median is Y3.

i.e. Y3 = m .

Now we want to find the probability that, P( Y2 < m < Y4 ) which is equal to the probability of P( Y2 < Y3 < Y4 ) , since , Y3 = m .

Now given that, Y1<Y2<Y3<Y4<Y5 be the order statistic. So Y3 must lie in between Y2 and Y4.

So , P( Y2 < Y3 < Y4 ) = P( Y2 < m < Y4 ) = 1 [Since, Probability of a sure event is one]