solution-old-exam1

# solution-old-exam1 - Stat 126 Exam 1 Spring 2006...

This preview shows pages 1–5. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Stat 126 Exam 1 Spring 2006 Solwl‘I‘W5 Name: ‘1 113.5%: I pledge that I have neither given nor received unauthorized aid on this exam. Signature: (1) Assume that events A and B are independent, events A and C" are mutually exclusive, and events B and O are also mutually exclusive. Moreover, P(A) : 1/2, P(B) = 1/3, and P(C) = 1/4. (a) Draw a Venn diagram. (b) Calculate F) B" (1 0°). "/3 (0) Calculate P(A U B U C). V [2. ((1) Calculate P(B or C (2) Jeff plays the following game. A fair die is roiled. If the number 011 the die is odd, then Jeff loses. If the number on the die is even, the die is roiled again. In that case Jeff wins if the second number matches the ﬁrst and loses otherwise. L5: Luge ' W: Nth/1 (a) Construct a tree diagram. , (b) Find the probability that Jeff wins. J.J—:,\— 219 )2 (c) Given that Jeff loses, what is the conditional probability that he loses after the die is rolled twice? H b “Warsaw-- If gt:- 7: (WWW r- a (PC £0894 dim M M 3’0“ [08% ) :- H/ [2. (3) Imagine that Detroit Pistons will play Dallas Mavericks in the next NBA Finals. The team that reaches 4 wins ﬁrst wili he the champion. Hence the series may ﬁnish in at least 4 games and at most 7 games. For instance, a scenario that the Pistons take the series in 6 games would be the Pistons win the 6th game, and win other 3 out of the ﬁrst 5 games. Assume the outcomes of different games are independent, and assume for each game the Pistons win with probability 0.6. in .— w l --— (a) What is the probability that the Pistons will win the series in 5 games? (ll—f”) 01,3. Mf- 0.5 ’53 0‘2‘073é 3 W M J‘h‘l a r W 3"” (b) What is the probability that the Mavericks will win the series? ‘ M A]: {Wvae/r.‘dgs wa'» the. mile.) in l it Li”)?! gal-7 ' my» ‘5 4‘ 3 5" 3 .1 Tale}? %P(A6)+KA7) I? OH‘ '1" (f 3 ) Onlf' -0.é - 0.46 '7‘" (3 )0.7- v.5 we + 043.043.041 2 (4) A EuroPean put option is a ﬁnancial derivative that gives the holder 3. right, but not an obligation, to sell the underlying stock at a pie-arranged exercise or strike price K. Suppose the stock price X N N(,u, 02) with ,u = \$100 and a = \$20. At the expiration of the put option contract, the option holder will sell the stock (i.e. exercise the Option) if X < K, or do nothing otherwise. Let K = \$110. (a) In theory, the stock price X should be positive. Check Whether the assumption “X ~ N (100,400)” is reasonable by calculating P(X > 0 . ) "(9000) swig?) 0%?) : M2 7-5) a: [ Haw I'M (WM-F “X40 U f4 mgli‘gi‘ék , (b) What is the probability that the option holder will exercise the option at its expiration? TOM/,0): /\’——n>o 4 {ID-400): Mg 4 0. y) : 0.69/5“. 10 1o (0) Suppose there are 60 put option contracts, each having the same strike price K = \$110 and same normal distribution as aforementioned for its associated stock. Assume the 60 associated stocks behave independently. Among the 60 put options, what is the probability that at most 40 of them will be exercised? Le+ y: a DP pwl' Opii'mis [W m 60 mews) fire/Fare averages , ' rue. . 7 Tim \//N Ei'nfép, MW?) ’ 40- éo°o.6?/5'+"'5“) Wm; 23 mg ,WWWyW; #037 g 5, “0.2.872 : /~e9.6(03 ’v é 35775 élﬂé) T’— 0.33??? (d) Suppose among the 60 put options given in (c), we know that 58 of them are exercised at their expiration dates. What is the conditional probability that the remaining two put options are not exercised? Ewe Mawcz , rt: mail/mar mi :5 7m, 3% M 7’"!— fwé ‘. be . 2 7 ( rm. rewlm’vj‘two pul 097%“ core M awaited) :1 [l « law/'3’) : 0.6752 72%“ ‘ (5a) Randomly choose a pair {m, n} from the set of 10 consecutive integers 11, 12, ..., 20. What is the probability that the difference between m and n is a nmltipie of 5‘? (5b) Randomly choose 6 distinct integers A = {(11, ..., as} from the set of 40 consecutive integers . 1, 2, ..., 40.. What is the probabiiity that some pair in A has a difference that is a multiple of . 5? -/ (71:4. ’Pwlméi'ii'ij 3W be. i 1%” W icoﬂwi'ﬁ MIR/W -. {loom 6m 7m. parade, but ctr?“ «25.404 me. 4ng“ in ewes {025},{taste/{5,33%}. AW? tws W in W Same wu'ﬂ’ (Larva Mia—t} Mite/Mme. as 61. WI‘?& 010 5‘ . ML» 7% ngm 6 mam (21%;) ﬁr swine— ?rw MM We at Emit 2 5% m M In mﬁsmwa. (6) Let Z ~ N (O, 1) be the standard normal random variable with density 45(2) :— ﬁ 5%”2, I Id («1 —00 < z < oo. Deﬁne X = 3Z + 1. 5WW (a) Calculate P(—1 < X < 4 [ 0 < X < 7). ’Fﬁldk‘ﬂ o'<)<<7)ﬂ ?(04X<9‘) :/ ff-rg-fcgéj) WWW) "‘ HOMO) New”) .... .. .é?3 WW3 (2 a 2 ) m “0.947% 2: 0.776w 0,7772,— (/« 042%) 0'60 s w .___. _. ._ (b) Find the limit of P(X > 1 +313 } IZI > t) as t “‘4 oo. / F(3‘}>t//%/>t)= 19(8 719([z-W) “(Wm 2&27L: New WW. Mew) T’(/€/7’6) iii) T. ,L 5 zﬂ-Z‘ﬂc) Z .Rw‘t>o '——u .——-— ...
View Full Document

{[ snackBarMessage ]}

### Page1 / 5

solution-old-exam1 - Stat 126 Exam 1 Spring 2006...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online