Acuarial Sociey of Idia EXAMINAIONS Jue 5 C4 (3) Models oal Marks - 5 Idicaive Soluio
Q. (i) a) Le U deoe he process described by 3 ad V deoe he process described by 4. he 5 e 5 PU [ ] PV [ ] ( e ).538! b) he arrivals of eiher rai will be Poisso wih parameer per hour. I half a hour he umber of arrivals will have a Poisso (3) disribuio. hus, PU [ + V 3] PU [ + V ] PU [ + V ] PU [ + V ] 3 3 3 e 3 e 3e.578! c) he waiig ime ill he e is epoeial wih parameer. I is irreleva wheher oe has jus missed a rai or o sice he epoeial disribuio has he memoryless propery. hus, he required probabiliy is: P e 4 >.3 4 d) he disribuio of he umber of 3 s i ime is Poisso (5). hus he probabiliy of eacly oe is 5 e 5. However, we do kow he ime ake for he 4 o arrive. he disribuio of his is epoeial wih parameer. hus, we codiio o his ime. As he ime is coiuous, we eed o use he desiy fucio: [ eacly oe 3 before a 4 arrives] [ eacly oe 3 i ime ] P P e d 5 5 e 5 de d (iegraed by pars) 3 (ii) As i every rais is o average a 4, he aswer is. Aleraively, le [] (iii) he waiig ime for a 3 be deoed by 3 ad ha for a 4 be deoed by 3. he probabiliy of a 4 arrivig before a 3 is he P[ 4 < 3]. Codiioig o he 4 waiig ime, we hus have 4 3 3 P[ < ] e P[ > ] d 5 e e d he disribuio of he umber of Chiaraja maufacured rais i ime i is 3 Poisso wih parameer + 5* 3. he waiig ime will be hus be epoeial wih parameer 3. Sice he epoeial disribuio has he lack of []
memory propery, ha a Chiaraja maufacured rai has come for over a hour is irreleva. he epeced waiig ime is hus 3 hours or abou 7.7 miues. oal [5] Q. A sochasic process X is saioary if he joi disribuios of he X,X,...,X ad X,X,...,X are ideical for all + k + k + k m m,,...,, k+, k+,..., k+ ad all iegers m. m m he process is weakly saioary of he epecaios E[ X ] are cosa wih respec o ad he covariaces Cov( X, X oly o he lag k. +k ) deped If ad +u are i he se of permissible values, he he icreme for ime u will be X+u X. For a discree process he Markov propery requires ha: P X X,X,...X P X m m X m m for all imes < <... < m < ad all saes < <... < m <. A discree ime marigale X saisfies wo codiios:. E[ X ] < for all. E[ X X,X,...X m] X m for all m <. Q.3 a) A each sep Aishwarya s fud will chage by a radom amou Z where Z + wih probabiliy oal [] Z wih probabiliy So ha S S + Z will be a simple symmeric radom walk. Iiially S k he boudary codiios are such ha. [ S ] [ ] PS ad PS K S K So ha we ca cosider i as a walk o (,,..K).
b) Here, E S S, S,..., S E S + Z S, S,..., S [ ] [ ] S E[ Z] S + if < S < K. I he case where S or S K, he above propery holds sice he S S. c) A soppig ime for he process S is a posiive valued ieger radom variable such ha for all, he idicaor variable { } if ad { } I I oherwise is a fucio of he pas ad prese values S, S,... S oly. he opimal soppig heorem saes ha E[ S ] E[ S ] if eiher. is bouded, i.e. N for some cosa N. S is bouded, i.e. S N for some cosa N d) Le be he soppig ime uil he process reaches or K. Sice S is a marigale ad bouded, he opimal soppig heorem ca be applied. his resuls i E[ S ] E[ S ] k. hus, ES [ ]. PS [ S k] + KPS. [ K S k] K( PS [ S k]) Sice he probabiliy of rui is jus PS [ S k], ad we kow K k ha ES [ ] k, we ca hus obai PS ( S k) K e) If Aishwarya is greedy he K is very large. If we ge K he, K k PS ( S k) K I oher words, she is cerai o be ruied if she does o qui. [] f) Vivek eeds o be super rich so ha he will o go broke. he soppig ime herefore oly depeds o Aishwarya s posiio. [] oal [5] [] [4] Q.4 a) b) B N(,) ad B N(,). hus, PB ( ) Φ ().59 PB ( ) Φ.4 B N(,) hus ( ) P[ Z > zb > z] + P[ Z > zb < z] P[ Z > zb > z] P[ B z] P Z>z,,, > z Now P( Z>z) Φ c) z z z 4 f ( z) Φ Φ e z >. z π E( X ) EX ( ) if is he soppig ime ad X is a marigale. B is a [4]
marigale. hus, EB ( ) EB ( ) for all. is he firs ime he process X his. hus B + µ EB ( ) E ( µ ) E ( ) µ B µ X. hus oal[] *******************