A company is reviewing tornado damage claims under a farm insurance policy. Let X be the portion of a claim representing damage to the house and let Y be the portion of the same claim representing damage to the rest of the property. The joint density function of X and Y is f(x,y)=6[1-(x+y)], x>0, y>0, x+y<1. Determine the probability that the portion of a claim representing damage to the house is less than 0.2.

Alexia Avila 2022-11-21 Answered
Tornado Damage Modeling
A company is reviewing tornado damage claims under a farm insurance policy. Let X be the portion of a claim representing damage to the house and let Y be the portion of the same claim representing damage to the rest of the property. The joint density function of X and Y is
f ( x , y ) = 6 [ 1 ( x + y ) ] x > 0 ,   y > 0 ,   x + y < 1 .
Determine the probability that the portion of a claim representing damage to the house is less than 0.2.
What I did was establish f x ( X ) = 3 6 x since X represents damage to the house. Then I integrated the function from 0 to 0.2 and I got an answer of 0.48, however, the actual answer is 0.488.
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Answers (1)

cenjene9gw
Answered 2022-11-22 Author has 13 answers
Step 1
Let us do it like you did, first finding the density of X. To do this, we "integrate out" y. So we want
0 1 x 6 ( 1 x y ) d y .
The integration goes from y = 0 to y = 1 x because of the condition x + y < 1. The integral is
6 ( y x y y 2 2 ) | 0 1 x .
Plug in. We get 6 ( 1 x x ( 1 x ) 1 2 ( 1 x ) 2 ) . This simplifies to 3 ( 1 x ) 2 .
Now integrate from x = 0 to x = 0.2. Equivalently, find the double integral of the joint density, y = 0 to 1 x, x = 0 to x = 0.2.
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