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Chapter 6: Q.6.10 (page 271)

ThejointprobabilitydensityfunctionofXandYisgivenbyf(x,y)=e鈭�(x+y)0鈥�x<q,0鈥�y<qFind(a)PX<Yand(b)PX<a

Short Answer

Expert verified

(a) P(X<Y)=12

(b)P(X<a)=1-e-a

Step by step solution

01

Introduction

The joint probability density function of X and Yf(x,y)=e鈭�(x+y)0鈥�x<q,0鈥�y<q

02

Explanation of (a).

Giveninformation:Thejointprobabilitydensityfunctionisf(x,y)=e-(x+y)鈮�x鈮�鈭�,0鈮�y鈮�鈭�formula used:P(X<Y)=鈭�0鈭�鈭�0yf(x,y)dxdy鈭�0鈭�鈭�0ye-x脳e-ydxdy鈭�0鈭�-e-x0y脳e-ydy鈭�0鈭�-e-y+1脳e-ydy鈭�0鈭�e-y-e-2ydySolvingitfurther-e-y-e-2y-20鈭�=1-12=12
03

Explanation of (b).

Formula used:
P(X<a)=鈭�0鈭�鈭�0af(x,y)dxdyCalculation:Findtherequiredprobabilityusingtheaboveformula鈭�0鈭�鈭�0ae-x脳e-ydxdy鈭�0鈭�-e-x0a脳e-ydy鈭�0鈭�-e-a+1脳e-ydy-e-a+1脳e-y-10鈭�=1-e-a

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