PASS_wk4_final

PASS_wk4_final - PASS for Actuarial Studies and Commerce...

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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 five-sided 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 anti-clockwise manner. Each day, Khar Mun moves either one step in the clockwise direction with probability p or one step in the anti-clockwise direction with probability 1-p. So for example, on the first day, Khar Mun will move to City 1 with probability p and City n with probability 1-p. 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=1-p). ) 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 ( ) ...
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