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Unformatted text preview: STAT 479
Test 1
Spring 2012 February 14, 2012 1. Losses during 2011 were distributed as a gamma distribution with a variance of 4,480,000. Losses are expected to be subject to P % inflation for 2012. The variance of losses for 2012 is
expected to be 17,920,000. Determine P. §\® ’ :22“ " @329); in; + Wag @169 ’ i b \30 0L 5 (faves. AQIE m Um a d (@293 30% at m@% )7, anﬂgoo 2: p/mié/l Légojooéb . The cost of one day in the hospital is distributed as a two point mixture distribution. The
distribution is 0.7 of a Pareto distribution with parameters of 6 = 5000 and or = 5 and 0.3 of a
Gamma distribution with a mean of 10,000 and a variance of 5 million. Hassan Hospital has 100 patients today. Assume the cost of each patient is independent of the
cost of any other patient. Using the normal approximation, calculate the probability that their total cost will exceed
500,000. ale 0% (iooooﬁaw ggvga km
‘S\
\lom 0 vi» 5%: (§;pm,Q©@ “l” ze/weﬁ 3. The number of vehicles that arrive at a gas station during an hour is distributed as a Poisson
distribution with a mean of 60. Of every 500 vehicles that arrive at the gas station, one is a
Toyota Prius. Calculate the probability that the number of Priuses that arrive at a gas station in a 24 hour
period will exceed the expected number of Priuses. W r \ A if“) a,
t: (Notme %?Qfllsgs rm kl 0mm)
:3?” (OOXQWQ%X%:%$ % (Pr (Number :7 a6} 3 4. The number ofdropped calls in a month on an iPhone is distributed as a Negative Binomial with
a mean of 8 and a variance of 40. N represents the number of dropped calls for any other cell phone which is distributed as a
Negative Binomial with the same parameters as the iPhone except that the probability of having
no dropped calls in a month is set to 20%. Calculate the Var[N]. 1%mz (are vain 3’5sz 5. The number of claims associated with a critical illness policy, represented by the random
variableN , is distributed as a Poisson distribution with a mean of A. Further, you are given
that l is distributed as an exponential distribution with a mean of 1.3. Calculate the Var Va 20: Ewan/13) +l/au [Em/x3] LB
Vmiﬁ‘thEXy l
l
l
galls 6. In 2011, losses were uniformly distributed between 0 and 50,000. An insurance policy pays for
all claims in excess of an franchise deductible of $5,000. For 2012, claims will be subject to
uniform inflation of 8%. The insurance company replaces the franchise deductible with an
ordinary deductible of d so that the expected cost per loss in 2011 is the same as the
expected cost per loss in 2012. Calculate d. @ 5X"; [0%,000 '5’ 2‘42]??? 9.9g, 4g :3 6:3 7. Losses represented by X are distributed as a Paretodistribution with parameters 9 = 60 and
a = 2. _ TVaR75 (X)
VaR75 (X) ' RX Losses represented by Y are distributed uniformly between 0 and 100. _ TVaRp (Y)
R’ _ VaRp (Y) ' Determine p such that RX =Ry. 8. You are given the following losses suffered by policyholders of Datsenka Dental Insurance
Company: 45, 50, SO, 50, 60, 75, 80, 120, 230 The random variableX represents the losses incurred by the policyholders. Treating this data
as an empirical distribution, calculate: a. E[X]
b. Var[X] c. The mode of the distribution cl. Datsenka has issued a dental policy that has an ordinary deductible of 50 and an upper
limit of 100. The upper limit is applied to the loss before the deductible. Calculate
Datsenka’s expected payment per payment. Lila? jigsaw” 51:.) M72 aw at; wages; is g exec; 3m , a. #2 & .laoga Lao £2?
2 Maize?) W yﬁ AM. if”;
a c Q  a [a ‘ we 35% em is»??? $3735 W”? ., m . f 90%? B0 im s3 2%ga9‘ﬁ13 . You are given the following empirical distribution of losses: 1000 4000 5000 6000 10,000 Wang insurance Company wants to apply an ordinary deductible to these claims so that the Loss
Elimination Ratio will be 0.30. Determine the deductible. Q or; "iv esﬁoo if» @QQQWﬁWQ m {a a? ﬂea W in o Cm‘lmﬁieﬁm, 10. You are given f(x) = 3x2
1000 for 0 s x S 10 . Calculate the median of this distribution. w ug m {Wm‘ng “:51” my \
3
B 2: “‘77 93557
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