This preview shows page 1. Sign up to view the full content.
Unformatted text preview: PASS for Actuarial Studies and Commerce
(ACTL1001)
2010 Session 2 – Week 4
Probability, Simulation and Gambler’s Ruin:
1. Chewbacca 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.
( ). Show that:
2. 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
and variance
3. A random claim has a lognormal distribution with
and
and the probability of obtaining a claim amount greater than
.
4.
is exponentially distributed with pdf ( )
(
).
(a) Derive the cdf ( ).
(b) Find ( ).
(c) Find Var(X) . . Find its expected value, 5. * Lando 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?
6. **There are cities arranged in a circle. Khar Mun is currently at City 0. Suppose that the cities are
labelled 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? (Hint: this is a gambler’s ruin problem.)
7. ***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
(
8. * Suppose that for 9. Using the results in 3, calculate )
( . Prove that
, given , ) , and ( ) Solutions to Exercises
Question 1
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 2
( )
( )
( () ) () () ( ) ()
()
() ( ) ()
[ ( )]
() () Question 3
( ) ()
( ) ( ) ( *
( ) Question 4
() ∫ () ∫ () ∫ Question 5
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 , we get:
( (
(
(
Therefore: )(
(
(
(
( )(
)(
)(
)( )
)
) )
)
)
)
) in terms of and . points. )
) (ie. go upwards from to & ( )( Adding them up:
( )[( ) (
)( )( (
( ) )( )( ( ( )] )) )) ( ( )
( So by letting
( ( ) )) (and so
) ): Question 6
Let A represent the event, every city visited by time of first return to city 0.
()
(
(
( )( )
)( ) )
( )( ) ) This probability can be thought of as
Consider ( 
a gambler’s ruin problem  we want the probability that Khar Mun reaches city n before he
reaches city 0. To put it into a more familiar framework, let Khar Mun start off with 1 wealth at
city 0. He moves clockwise. Let every move clockwise represent a win (i.e. he gains one wealth),
and every move anticlockwise represent a loss (loss of one wealth). We want the probability
that he reaches N wealth (reaches city N) before he reaches 0 wealth (reaches city 0 again). This
is the solution to the gambler’s ruin problem, with probability of success p. ( ) Likewise,
( ) ( (the solution to the gambler’s ruin problem with probability of success q=1p). ) So () ( ( ) ) ( )( ( ) ) Question 7
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:
( ⋃ ( Question 8 ) ⋃
) ( ∑ (( ) *+ ), ( ( ) Question 9 ( ) ...
View Full
Document
 One '09
 Nicole

Click to edit the document details