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Test1-S08-Solutions

Test1-S08-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) The loss random variable X has pdf f X ( x ) = 1 4 (3 + x ) e & x ; x > . (a) (2 marks) Determine the cumulative distribution function F X ( x ) . F X ( x ) = Z x 1 4 (3 + y ) e & y dy = 1 4 & & (3 + y ) e & y ¡ ¡ x + Z x e & y dy ¢ = 1 4 £ 3 & (3 + x ) e & x + ¤ 1 & e & x ¥¦ = 1 4 £ 4 & (4 + x ) e & x ¦ = 1 & 4 + x 4 e & x . (b) (1 mark) Determine the hazard rate h X ( x ) . h X ( x ) = f X ( x ) 1 & F X ( x ) = 3+ x 4 e & x 4+ x 4 e & x = 3 + x 4 + x = 1 & 1 4 + x . (c) (2 mark) Is the random variable X DFR, IFR or neither? Justify your answer. h X ( x ) is a non-decreasing function of x ) X is IFR 1 (d) (2 marks) Determine the mean excess loss e X ( x ) = E [ X & x j X > x ] . e X ( x ) = E [ X & x j X > x ] = Z 1 x ( y & x ) f X ( y ) F X ( x ) dy = Z 1 x ( y & x ) 3+ y 4 e & y 4+ x 4 e & x dy = Z 1 x ( y & x ) (3 + y ) e & y (4 + x ) e & x dy = 1 (4 + x ) e & x Z 1 x ( y & x ) (3 + y ) e & y dy = 1 (4 + x ) e & x Z 1 x ( y & x ) (3 +...
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Test1-S08-Solutions - ACTSC 431 Loss Models 1 TEST#1 1(13...

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