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In: Statistics and Probability

If T1 and T2 are independent exponential random variables, find the density function of R=T(2) -...

If T₁ and T₂ are independent exponential random variables, find the density function of R=T₍₂₎ - T₍₁₎.

This is for the difference of the order statistics not of the variables, i.e. we are not looking for T₂ - T₁. It is implied that they are both from the same distribution. I know that

f_T(t) = λe^-^λt

f_T(1)T(2)(t₁,t₂) = 2 f_T(t₁)f_T(t₂) = 2λ² e^-λt₁ e^-λt₂ , 0 < t₁ < t₂ and I need to find f_R(r).

From Mathematical Statistics and Data Analysis by Rice, 3rd Edition, Question 79 from Chapter 3. The solution in the back of the book: "Exponential (λ)". I am not sure how to get to that answer.

Solutions

Expert Solution

let be random samples from exponential distribution with

pdf

cdf

Using the formula for joint pdf of

if are the ordered statistics, the joint pdf is

Let and giving us 0<R<S

That is can be expressed in terms of S and R as

The Jacobian |J| is

the joint pdf of R,S is

The marginal pdf of R is

This is the pdf of exponential distribution with parameter

That means

orchestra answered 10 months ago

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