In Example 6.14, Y₁ and Y₂ were independent exponentially distributed random variables, both with mean B. We defined U₁=Y₁/(Y₁+Y₂) and U₂ = Y₁+ Y₂ and determined the joint density of (U₁, U₂) to be e-42/A₂ 0

Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter13: Probability And Calculus
Section13.CR: Chapter 13 Review
Problem 42CR
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*6.63 In Example 6.14, Y₁ and Y₂ were independent exponentially distributed random variables, both
with mean B. We defined U₁ = Y₁/(Y₁ + Y₂) and U₂ = Y₁+ Y₂ and determined the joint
density of (U₁, U₂) to be
0₁.0₂(₁,₂)=242€¯
0,
0<u₁ <1, 0< u₂,
otherwise.
Show that U₁ is uniformly distributed over the interval (0, 1).
a
b
Show that U₂ has a gamma density with parameters α = 2 and B.
c Establish that U₁ and U₂ are independent.
Transcribed Image Text:*6.63 In Example 6.14, Y₁ and Y₂ were independent exponentially distributed random variables, both with mean B. We defined U₁ = Y₁/(Y₁ + Y₂) and U₂ = Y₁+ Y₂ and determined the joint density of (U₁, U₂) to be 0₁.0₂(₁,₂)=242€¯ 0, 0<u₁ <1, 0< u₂, otherwise. Show that U₁ is uniformly distributed over the interval (0, 1). a b Show that U₂ has a gamma density with parameters α = 2 and B. c Establish that U₁ and U₂ are independent.
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