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Suppose X₁ and X₂ have the joint par f(x1, x₂) fw (w) For constants w₁ > 0 and w₂ > 0, let W = W₁X₁ + W₂X2. (a) Show that the pdf of W is = { 0 = elsewhere. е-x¹е-x2 { x1 > 0, x₂ > 0 elsewhere. W1-W2 0 1 · (e-w/w₁ 1- e-w/w₂) w > else (b) Verify that fw(w) > 0 for w > 0. (c) Note that the pdf fw(w) has an indeterminate form when w₁ = W₂. Rewrite fw(w) using h defined as w₁ - W₂ = h. Then use l'Hôpital's rule to show that when w₁ = W₂, the pdf is given by fw(w) = (w/w) exp{-w/w₁} for w> 0 and zero

Suppose X₁ and X₂ have the joint par f(x1, x₂) fw (w) For constants w₁ > 0 and w₂ > 0, let W = W₁X₁ + W₂X2. (a) Show that the pdf of W is = { 0 = elsewhere. е-x¹е-x2 { x1 > 0, x₂ > 0 elsewhere. W1-W2 0 1 · (e-w/w₁ 1- e-w/w₂) w > else (b) Verify that fw(w) > 0 for w > 0. (c) Note that the pdf fw(w) has an indeterminate form when w₁ = W₂. Rewrite fw(w) using h defined as w₁ - W₂ = h. Then use l'Hôpital's rule to show that when w₁ = W₂, the pdf is given by fw(w) = (w/w) exp{-w/w₁} for w> 0 and zero

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Suppose X₁ and X₂ have the joint par ƒ(x1, x2) = { 8 0 For constants w₁ >0 and w₂ > 0, let W = W₁X₁ + W₂X₂. (a) Show that the pdf of W is е-x¹е-x2 1 fw (w) W1-W2 0 { = elsewhere. x1 > 0, x₂ > 0 elsewhere. 1- e-w/w₂) w > else · (e-w/w₁ (b) Verify that fw(w) > 0 for w > 0. (c) Note that the pdf fw(w) has an indeterminate form when w₁ = W₂. Rewrite fw(w) using h defined as w₁ - W₂ = h. Then use l'Hôpital's rule to show that when w₁ = W₂, the pdf is given by fw(w) = (w/w) exp{-w/w₁} for w> 0 and zero
Transcribed Image Text:Suppose X₁ and X₂ have the joint par ƒ(x1, x2) = { 8 0 For constants w₁ >0 and w₂ > 0, let W = W₁X₁ + W₂X₂. (a) Show that the pdf of W is е-x¹е-x2 1 fw (w) W1-W2 0 { = elsewhere. x1 > 0, x₂ > 0 elsewhere. 1- e-w/w₂) w > else · (e-w/w₁ (b) Verify that fw(w) > 0 for w > 0. (c) Note that the pdf fw(w) has an indeterminate form when w₁ = W₂. Rewrite fw(w) using h defined as w₁ - W₂ = h. Then use l'Hôpital's rule to show that when w₁ = W₂, the pdf is given by fw(w) = (w/w) exp{-w/w₁} for w> 0 and zero
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