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Test1-F09-Solutions

Test1-F09-Solutions - ACTSC 431 Loss Models 1 TEST#1 1(13...

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Unformatted text preview: ACTSC 431 - Loss Models 1 TEST #1 1. (13 marks) Let X be a random variable with cumulative distribution function F X ( x ) = & x & ¡ & , < x < & , for ¡ > 1 . De&ne Y = 1 X & 1 ¡ . (a) (3 marks) Find the density function of Y . Solution: By de&nition, F Y ( y ) = Pr ¢ 1 X & 1 & ¡ y £ = Pr ¢ 1 X ¡ y + 1 & £ = Pr X > 1 y + 1 ¡ ! = 1 & F X ¢ & &y + 1 £ = 1 & ¢ 1 1 + &y £ & , for y > . It follows that f Y ( y ) = &¡ (1 + &y ) & +1 , y > . (b) (3 marks) Using (a), show that the distribution of Y is a scale distribution. Determine if the distrib- ution of Y has a scale parameter and identify this parameter (if it exists). Solution: Let Z = cY . As shown in class, the density function of Z is given by f Z ( z ) = 1 c f Y & z c ¡ = 1 c &¡ ¤ 1 + & z c ¥ & +1 = & & ¡ (1 + & & z ) & +1 , for z > . Clearly, the distribution of Z is of the same parametric form as Y . As a result, one concludes that the distribution of Y is a scale distribution....
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Test1-F09-Solutions - ACTSC 431 Loss Models 1 TEST#1 1(13...

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