Question

solve d, e, f

ASP please

Let Y denote the claim size for a certain type of insurance policy and suppose Y has an exponential distribution with mean $300,000. (a) Write the density of Y for all values of y. (b) Determine the probability that a claim is smaller than $400,000 Define X = -Y so that X gives claim sizes that are negative, representing a payout from the insurance company. The remaining parts of this problem involve analyzing the random variable X. You may find it easier to substitute X = -Y and work with the random variable Y and its exponential distribution/density. (c) One measure of the risk of an insurance policy is the "Value at Risk”. For 0 < p < 1, the Value at Risk, VaR(p), is defined to be the value to such that proportion p of the claims are worse than xo: P(X <30) = p. Show that for p = 0.05, 30 = VaR(0.05) = -$300,000 In (20) -$898,719.68. (d) Let to = VaR(0.05) = -$300,000 ln(20) from (c). Determine the conditional distribution function F( X<Xo) for all values of x: the conditional distributon function is define to be F(x|X<20) = P(X<x|X<20), -o0<x< . (e) Determine the conditional density f(x|X<20) of X given that X < Xo for all values of x. Hint: Think about the relationships between a distribution function and its density function.] (f) Another measure of the risk of the insurance policy is the "conditional value at risk" or "tail expectation". It is defined for 0 < p < 1 to be the conditional expectation of X given X < VaR(p); it gives the expected claim size condi- tioned on those claims being worse than VaR(p). Again with Xo = VaR(0.05) = -$300,000 In(20), determine (to the nearest cent) PIE CVAR(0.05) = E[X|X < 20) = f(x|X < xo) dr. Jr=-00

Answer 1

Yourfo000) Y~exp 300000 300000 300000 eusewhere XE[0,-00) :- fxcx)= P(X<x) P(-Y<x) PCY>-X) 2, 300000 300000 300000 300000 300000 -160- 300000/ 300000 facas= dx Now, P(X<x0)=P fcxo)

Ajantel Page No. Data e3000 00 30000olmcod08 100 20 300000 1ncR0) F (x/X<X0) = P (X <x L X<Xo) PCX<X and X <xo) PCX<x0) PCX<x) FCRIX<) facta)

Page No. Dats e300000 0.05 0000 30 e 30o alhere 89871g 68. fextxexo) 200000 300 008 elsewhere feelx<x) 300000 15000 つてくズ。 elgewhere 3)

Paga No Date Cvar Co.05) fefe eftxxo) dx 300000 15000 %3D xe300000 to 15000 300000 to e 300000 300000 (xo e30000. 30000 1500 (300000)² 30000o 300000 ποου 300000 15000 300 000 C300000) x0.05] 3000 -1198 3719.68