Consider two continuous random variables X and Y with joint density function

jernplate8

jernplate8

Answered question

2021-10-26

Consider two continuous random variables X and Y with joint density function
f(x,y)=beg{cases}x+y ox1,0y10    otherwiseend{cases}
P(X>0.8, Y>0.8) is?

Answer & Explanation

Szeteib

Szeteib

Skilled2021-10-27Added 102 answers

Step 1 
Given: 
The joint density function for the two continuous random variables X and Y is defined as:
f(x,y)=beg{cases}x+y ox1,0y10    otherwiseend{cases} 
Then, 
P(X>x,Y>y)=y1x1f(x,y) dx  dy  
Step 2 
P(X>0.8,Y>0.8)=y=0.81x=0.81(x+y) dx  dy  
=y=0.81{x=0.81(x+y) dx } dy  
=y=0.81[x22+xy]0.81 dy  
=y=0.81{(12+y)(0.822+0.8y)} dy  
=y=0.81{12+y0.320.8} dy  
=y=0.81(0.18+0.2y) dy  
=[0.18y+0.2y22]0.81 
=[(0.18+0.1)(0.18×0.8+0.1×0.82)] 
=0.072 
The required probability is 0.072.

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