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Unformatted text preview: PASS for Actuarial Studies and Commerce (ACTL1001)
2008 Session 2 – Week 3
Probability
1. Intel sells quadcore processors for $2000. The life time of each processor (in years) has an exponential
distribution with a cumulative distribution function given by:
If a processor fails in the first year, Intel refunds half the cost. If it fails in the second year, Intel refunds one fifth
of the cost. What is the expected amount refunded over the next 2 years for each processor sold?
2. AMD manufactures quadcore processors with a failure probability of 0.25. In a random sample of 10 processors,
what is the probability that there are exactly two defective processors?
3. * Selina, Hebe, and Ella each toss a coin. The winner of the game is the one who has a different coin toss than
the others. If all of them have the same coin toss, then they repeat until there is a winner. After playing 6 games,
what is the probability that Selina loses at least 5 times?
4. The IAA Part I CT4 exemption requires UNSW students to pass both ACTL2003 and ACTL3001. If a candidate fails
in either subject, he need only repeat that subject. The probability of passing ACTL2003 for two candidates
Kerrigan and Zeratul are both , and their probabilities of passing ACTL3001 are both . Find the probability that:
(a) Kerrigan obtains the exemption by attempting each subject only once.
(b) Kerrigan fails in one of the two subjects, but obtains the exemption by repeating once.
(c) At least one of Kerrigan and Zeratul obtains the exemption on the first attempt.
5. * Bernard draws 3 balls from a bag containing
at least one red ball as . Find
. red balls and blue balls. Denote the probability of obtaining 6. * A real number is chosen with equal probability from the interval
chosen from
. What is the probability that is larger? . Another real number is similarly 7. Of the following, which has the greatest probability?
(a) Obtaining exactly 1 six in 4 tosses of a die.
(b) Obtaining exactly 2 sixes in 8 tosses of a die.
(c) Obtaining exactly 3 sixes in 12 tosses of a die.
8. is exponentially distributed with pdf
(a) Derive the cdf
.
(b) Find
.
(c) Find
. 9. * You are given that
expectation and variance of and
. . has a density . and are independent. Find the 10. Mike tosses an unbiased coin times. What is the probability
(note that 0 is an even number)
11. *Let
the pdf of that he will toss an even number of heads? be independent exponential random variables that have distribution { . Find (Hint: find the cdf of Y, then differentiate to find the pdf) 12. Changki plays a game which involves tossing a fivesided die (with numbers 1,2,…,5) 1000 times. The payoff for
playing this game is the total of these 1000 tosses.
(a) Simulate the game 10 times using EXCEL (or some other software). Use your results to estimate the expected
payoff.
(b) Calculate the expected payoff theoretically.
13. You are given that is Normal with mean and variance . Let
(a)
(b)
Note that for any constant ,
will also be Normal and will have mean
14. A random claim has a lognormal distribution with
probability of obtaining a claim amount greater than and . Show that: and variance . . Find its expected value, and the . 15. * Sachi plays a game in which he scores 2 for a success (probability ) and 1 for a failure (probability
His current score is . What is the probability that his score will at some stage be exactly 100?
16. **There are cities arranged in a circle. Khar Mun is currently at City 0. Suppose that the cities are labeled in
ascending order in an anticlockwise manner. Each day, Khar Mun moves either one step in the clockwise
direction with probability p or one step in the anticlockwise direction with probability 1p. So for example, on
the first day, Khar Mun will move to City 1 with probability p and City n with probability 1p. What is the
probability that Khar Mun will have visited every city by the time he first returns to City 0?
17. ***There are n actuarial students at Jamie’s Starcraft 2 LAN Party. Suppose that all students decide to throw
their computer mice into the center of the room. Each student then randomly selects a mouse. Show that the
probability that none of the students selects their own mouse is ). Solutions to Exercises
Question 1
[ Question 2
( ) Question 3 (* ( )( * ( * Question 4
(a)
(b) ( ) ( ) (c) Question 5
( *( *(
( [
( * *( *
*( *] Question 6 ( *( * ( *( * ( *( * Question 7
( )( * ( *
( )( * ( *
( )( * ( * Therefore option (a) has the highest probability. Question 8
∫
∫
∫ Question 9
∫ ∫ Question 10
Construct a recurrence relation based on the outcome of the next toss: Question 11
The cdf of Y is defined as the following: Due to independence, and since all the random variables are identically distributed, this is equal to
[ (∫ ) Using the relation between cdfs and pdfs, we find that Question 12
An example excel worksheet (for one play of the game) is: A
(Generated Random Number) B
(Generated Roll of Die) =RAND()
=RAND()
…
=RAND() =CEILING(5*A1,1)
=CEILING(5*A2,1)
…
=CEILING(5*A1000,1)
=SUM(B1:B1000) 1
2
…
1000
1001
Question 13 [ Question 14 ( * Question 15
In each game, he either succeeds or fails, so
When his score is , let the probability that it will at some stage be exactly 100 be
When his score is , if he wins the next game, then he will have
If he loses the next game, then he will have
points.
So Now, we know that
and
So we want to use the recurrence to get
From in terms of and , we get: Therefore: Adding them up:
[ ( (
( ) ) ) So by letting
( (and so
) ): . points. (ie. go upwards from to & ). Question 16
Let A represent the event, every city visited by time of first return to city 0. Consider
This probability can be thought of as a gambler’s
ruin problem  we want the probability that Khar Mun reaches city n before he reaches city 0. From lecture
notes, the probability for this, given that probability p represents the probability he moves up one city, and
imagining that starting at city 0 is equivalent to a gambler starting with wealth 1, with a movement up a city
representing a win, the probability is Likewise, So Question 17
Let denote the event that actuarial student gets their own mouse. So the probability that no actuarial student
gets their own mouse is
( ⋃ (∑ ⋃ )
∑ ( ⋂ ⋂ Number of terms in the summation ∑ ⋂ is ( ). ⋂ ⋂ ), ∑ ⋂ ⋂ () Generalizing this idea, and applying it leads to:
( ⋃ ⋃ ) (∑ ( *+ ...
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