Question

In: Statistics and Probability

2. Let X1, X2, . . . , Xn be independent, uniformly distributed random variables on...

2. Let X1, X2, . . . , Xn be independent, uniformly distributed random variables on the interval [0, θ]. (a) Find the pdf of X(j) , the j th order statistic. (b) Use the result from (a) to find E(X(j)). (c) Use the result from (b) to find E(X(j)−X(j−1)), the mean difference between two successive order statistics. (d) Suppose that n = 10, and X1, . . . , X10 represents the waiting times that the n = 10 people must wait at a bus stop for their bus to arrive. Interpret the result of (c) in the context of this scenario.

Expert Solution

A bit theory

Suppose that denotes a random sample of size m from a continuous pdf f(y) and cdf F(y), where f(y)>0 for , a<y<b. Then the pdf of i th order statistic, be,

for and 0 otherwise.

Solution

Now here be independent, uniformly distributed random variables on the interval with pdf,

And the cdf as,

(a) So the pdf of j th order statistic be,

(b) Now we have,

, assuming and as

(c) So here we have,

(d) So here we suppose that n=10, and represent the waiting times that the n=10 people must wait at a bus stop for their bus to arrive. Then represents j th order waiting time where that person arrived at the bus stop after n-j people. So here we note that the waiting time difference of j th and j-1 th person doesnot depend on the value of j hence it is independent if the ordering of people according to first come first one to board in the bus basis.

Hence the answer...............

Thank you...........

orchestra answered 2 years ago

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