Test2_SOLUTIONS_F23_STAT230 (1)

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STAT 230 Test 2 University of Waterloo Question 1 2 3 Total Points 10 8 12 30 Instructions 1. Your final numerical answers should be given to 3 decimal places (e.g. 23.456 or 0.0000456). However, between steps/parts you should carry more decimal places to avoid rounding errors. 2. If working is not specifically requested, a correct answer will receive full marks. However, an incorrect answer could still receive credit if working is provided. An incorrect answer with no working will receive zero marks. 3. The exam is 50 minutes. The number of marks available per question is indicated in [square brackets]. 4. Answer the questions in the spaces provided. You may use the last page of the test for any additional work you would like to be graded. If you use this space, make it as clear as possible which question(s) your work relates to. 5. A formula sheet of discrete distributions is provided at the back of the exam. 6. Only non-graphing non-programmable calculators are permitted. University of Waterloo STAT 230 – Tutorial Test 2 – Fall 2023 Page 1 of 6

1. Driving across the Quebec countryside, Audrey gets bored and starts counting the moose she sees on her never- ending car ride. Suppose that moose sightings follow a Poisson process with an average of 12 moose an hour on this cross-province journey. (a) [2] Denote by the random variable X the number of moose Audrey will see in the next t hours. Write down the probability function of X . For full credit, please state the range of the random variable. Sol: If X denotes the number of moose Audrey sees in t hours, then X ∼ Poi ( λt ) where λ = 12 (moose per hour). Hence, f ( x ) = P ( X = x ) = e − 12 t (12 t ) x x ! , x = 0 , 1 , 2 , . . . (b) [2] How many moose can Audrey expect to see during an 8 hour drive? Sol: The number of moose Audrey sees during an 8 hour drive, say Z , satisfies Z ∼ Poi (12 · 8) = 96 with E ( Z ) = 96 . Thus, Audrey can expect to see 96 moose on an 8 hour ride. (c) [3] Compute the probability that Audrey will see fewer than three moose in the next 30 minutes. Sol: If Y is the number of moose Audrey will see in the next 30 minutes, then Y ∼ Poi (12 · 0 . 5) or Y ∼ Poi (6) . Then P ( Y < 3) = P ( Y = 0) + P ( Y = 1) + P ( Y = 2) = e − 6 ( 6 0 / 0! + 6 1 / 1! + 6 2 / 2! ) = 0 . 0620 (d) [3] Now suppose Audrey stops every 30 minutes for the next 5 hours, thus creating 10 distinct, non-overlapping 30-minute segments. What is the probability she sees fewer than 3 moose on exactly 2 of these segments? Sol: From b) we know the probability of seeing fewer than 3 moose in a 30-minute interval is 0.0620. If W denotes the number of time intervals Audrey sees fewer than 3 moose, then W ∼ Bin (10 , 0 . 062) and we get P ( W = 2) = 10 2 0 . 062 2 (1 − 0 . 062) 10 − 2 = 0 . 104 University of Waterloo STAT 230 – Tutorial Test 2 – Fall 2023 Page 2 of 6

2. The OLG scratch and win game “Roulette" printed 2,500,000 tickets. These tickets sold for $2 each at varying retailers. Of these, the following prizes were given out and the following frequency. Prize ($) 35,000 1000 100 20 10 8 4 2 Frequency 3 7 100 11,000 2500 93,000 150,000 290,000 (a) [3] Find the net winnings of a random OLG roulette ticket. Sol: QQQQQQQ xf ( x ) = 105 , 000 + 7000 + 10 , 000 + 220 , 000 + 25 , 000 + 744 , 000 + 600 , 000 + 580 , 000 = 2 , 291 , 000 1 N QQQQQQQ xf ( x ) = 2 , 291 , 000 / 2 , 500 , 000 = 0 . 9164 Net winnings = 1 N QQQQQQQ xf ( x ) − 2 = − 1 . 0836 (b) [3] Find the net winnings of a random OLG roulette ticket that is NOT one of the largest 10 prize winners. Sol: QQQQQQQ xf ( x ) = 2 , 291 , 000 − 105 , 000 − 7000 = 2 , 179 , 000 1 N QQQQQQQ xf ( x ) = 2 , 179 , 000 / 2 , 500 , 000 = 0 . 8716 Net winnings = 1 N QQQQQQQ xf ( x ) − 2 = − 1 . 128 (c) [2] Find the net winnings of a winning OLG roulette ticket (that is, any ticket that wins something). Sol: N win QQQQQQQ n i = 3 + 7 + 100 + 11000 + 2500 + 93000 + 150000 + 290000 = 546 , 610 1 N win QQQQQQQ xf ( x ) = 2 , 291 , 000 / 546 , 610 = 4 . 1913 Net winnings = 1 N win QQQQQQQ xf ( x ) − 2 = 2 . 191 University of Waterloo STAT 230 – Tutorial Test 2 – Fall 2023 Page 3 of 6

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3. Gerry and his sister, Maddie, play a game in which they both select an integer between 0 and 21 (inclusive) at random. If Gerry’s and Maddie’s integers match, they score 22 points, otherwise they score 0 point. (a) [3] What is the probability that Gerry’s integer is at least x ? Please provide your answer as a function of x , defined for all x ∈ [0 , 21] . Sol: Let X ∼ Unif (0 , 21) be Gerry’s integer. Then P ( X ≥ x ) = 1 − P ( X ≤ x − 1) = 1 − F ( x − 1) = 1 − [ x − 1] − 0+1 21 − 0+1 = 1 − [ x ] 22 . (b) [4] Suppose Gerry and Maddie love this game and have played it 1,000 times. Calculate the expected value and the standard deviation of the total number points they scored. Sol: Let W ∼ Bin (1000 , 1 / 22) be the number of games with a match. E (22 W ) = 22(1000)(1 / 22) = 1000 points. SD (22 W ) = qqqqqqq 22 2 (1000)(1 / 22)(21 / 22) = √ 21000 = 144 . 914 points. (c) [3] Suppose Gerry and Maddie love this game and have played it 1,000 times. Approximate the probability that they scored a total of 990 points. Sol: Let M ∼ Bin (1000 , 1 / 22) ≈ Poi (1000 / 22) be the number of games with a match. We want P ( M = 990 / 22) = P ( M = 45) ≈ e − 1000 / 22 (1000 / 22) 45 45! = 0 . 0592 . (d) [2] Suppose Gerry and Maddie play the game until they score at least 100 points. Calculate the expected value of the number of games they play. Sol: Let Y ∼ NB (5 , 1 / 22) be the number of games without matches until 5 matches (110 points). E ( Y + 5) = 5 1 − 1 / 22 1 / 22 + 5 = 5(21) + 5 = 110 games. University of Waterloo STAT 230 – Tutorial Test 2 – Fall 2023 Page 4 of 6

Use this page for any additional work you would like to be graded. If you use this space, make it as clear as possible which question(s) your work relates to. If there is any ambiguity only your work on the previous question pages will be graded. University of Waterloo STAT 230 – Tutorial Test 2 – Fall 2023 Page 5 of 6

University of Waterloo STAT 230 – Tutorial Test 2 – Fall 2023 Page 6 of 6

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