Midterm1Solutions

Midterm1Solutions - ACTSC 431 Midterm # 1 FALL 2010 Name:...

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Unformatted text preview: ACTSC 431 Midterm # 1 FALL 2010 Name: ID 1. Suppose that the following probability density function of a random variable X is given: f<>~1(1 1)? 0< <@ “73 h 100 300 ’ “T u—P 500? 5 (a) Determine the cdf F = Pr(X g x) - o< .. Fm :[ /0L/7 (’“Eéif C23 ‘0 X i 2. = —f 3(/~5;§75 0 o (wit) i he : /~(/*sTO;)3 ov<><<§oo QD/@ 5 (b) Determine the limited expected value E (X /\ [ftX/bc.) jiffihflolt C9 5 ((3) Determine the failure rate 5 (d) Determine the mean residual lifetime 6(50). E(X~X)+ em) “1 ¥ FUX) V EX ' 500W) FM) 529 , if? 4 \ £9 : 4 ("500 ) J Q» 5 (Pg—é) _ éZoFOC’ng‘iq F 9 W325?) \ sow-x ‘ 4 5 (e) Determine the equilibriggdf Fe Fem” * QCOC) : if ( 60 94-0) M >230 q, : a» 22> kQUX) 3 U“ .3359) ® Em : (—7 (Iv figfi 5 5 (f) Determine Varp(X) with p = .95. ( )r 0‘ 5 Fx ? A g / ,, (fw r.” s: I r-Boo(/*30.0§ @’ : @7543 Mfr“) 187‘?” (D 2. The loss random variable X has cdf FX(:1:) = 1 — (1 +.013:)e_'01$, a: > 0. An ordinary deductible of 50, a loss limit of 200, and a coinsurance of 80% are applied to X. 5 (a) Determine the limited expected value function E (X /\ w03905 ~o.ol’X fiw) : o‘ol ( 1+ o.o\<x)e — oiole : mop-(xerox?le 7O er GW(2,/oo) {baa/9:400). \(o‘mlma/QL; (% _ O \omx , Like“ ‘9‘] 4 I~(’+o.o(o<>e ] -.0.ol9( :100[2'(2+0.0”X)Q J / 1 >0 @ 5 (b) Determine the cumulative distribution function of YP, the amount of payment given that a nonzero payment has been made. \ OJ Cofia'mkmw VFX(0() E (a) : (P “3 /v RM) _ .0 OH . \o.ovo( b [,‘(Howgmweo “3+ j -[l~(z+oro;d)e j CD ‘ I v 1+o.o\(y+of) -o‘oiovt ‘ I+O.o[c€ e v 0‘0! ~0i013/ x 1/ mee to => 55”“ = rwri+7%>6°‘°""'} “51””? ® 0118079. I tog/f : ugh? @ : :_ l... ,+Efié> efOfliZyg 5 (C) Determine E (YL) where YL is the amorint of payment on the loss. : ,,. 00,4 ‘ [Do [02* (2+o,o‘i)€ 2347’?) = can:me ‘E()</\d)j :7 0% ’00 [.2 ‘P.+»o‘u\\k)_e‘o‘o } MAJ erPcf J) r 6%{400 4 (2+ acme?” “+ r w 0-0; 3% 3e. 3' g’o K[’(.2+0\°’KZOO§¢:L+ (62+o.§>€‘5'§j ii (9 3. Let X , given that 9 = (9, have the conditional pdf f = 26’me‘9’fx2w > O. The pdf and mgf of 9 are given by fem) : \/27rfia3e_2%(9_;&)2, 6 > 0 and .. E (eta) = e;E (1— 1—%t>, t < (2M2) . 5 (a) Determine the conditional Cdf F = Pr (X [9 = 9) e (9 2 F049) 3 X079th ’6 Jt <9 0 -99 7‘ ""8 C9 0 I l — €ng 9( >0 5 (b) Determine the marginal pdf f of X. ...
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Midterm1Solutions - ACTSC 431 Midterm # 1 FALL 2010 Name:...

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