Question

Let Y1 < Y2 < Y3 < Y4 < Y5 be the order statistics of five independent observations from an exponential distribution that has a mean of θ = 3. (a) Find the p.d.f. of the sample median Y3. (b) Compute the probability that Y4 is less than 5.

Answer 1

Answer:

a) $10[1-e^{-y/3} ] ^{2} e^{-y}$

b)    ≈ 0.76

Step-by-step explanation:

Given that : Y1 < Y2 < Y3 < Y4 < Y5   are the order statistics of five independent observations

mean of θ  = 3

a) Determine the P.d.f of the sample median

P.d.f of sample median ( y3 ) = $10[1-e^{-y/3} ] ^{2} e^{-y}$    

attached below is the detailed solution

b) determine the probability that Y4 is < 5

p( Y4 < 5 ) = G4( 5 )

                 = 0.7599 ≈ 0.76

attached below is the detailed solutions