practice_final_prob_solutions

practice_final_prob_solutions - SDS 321 Practice questions...

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SDS 321: Practice questions 1. How many unique combinations can you get by rearranging the letters MISSISSIPPI? 11 letters so 11! permutations. 4 Is, 4Ss, 2 Ps, so 11! / (4!4!2!) unique combinations. 2. On the first day of a non-leap-year, I put $1 in a box. On the second day, I put $2 in the box. On the third day, I put $3 in. And so on. At the end of the year (365 days), how much money is in the box? The first day you put in $1, the last day you put in $365. The average of these two is 183. The average of the second and penultimate days is also 183. Etc. So, the total is 365 × 183 = $66795. 3. Let X be a normal random variable with mean 3 and variance 1, and let Y be a normal random variable with mean 4 and variance 2. (a) What is the distribution of Z = X + Y ? Normal (7 , 3) (b) What is the probability that Z is between 6 and 8? P (6 Z 8) = P ( 6 - 7 3 Z - 7 3 8 - 7 3 ) = P ( - 0 . 577 Z - 7 3 0 . 577) = 0 . 44 (from standard normal tables) 4. I am waiting for a bus, that I know will arrive at some time between 1pm and 2pm, with all times being equally likely. It gets to 1:30, and the bus has still not arrived. What is the probability that it arrives before 1:40? 1/3 5. Let X be a continuous random variable with PDF f X ( x ) = ( 0 . 125 x + 0 . 125 - 1 x 3 0 otherwise What is the PDF of Z = | X | ? f Z ( z ) = ( 0 . 25 0 x 1 0 . 125 + 0 . 125 x 1 < x 3 6. Alice and Bob are playing rock-paper-scissors. If both Alice and Bob play the same hand, they play again. What is the expected number of turns before someone wins? The time until someone wins is a Geometric distribution with p = 2 / 3 (probability someone wins). The expected value of a geometric distribution is 1 /p = 3 / 2. So, the expected number of goes before someone wins is 1 / 2. 1
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7. On a given day, a Poisson (100) number of insects fly through my yard. Using an appropriate approximation, what is the the probability that, over the month of May (31 days), the average number of insects is between 98 and 102? You may use the fact that a Poisson ( λ ) random variable has mean and variance λ .
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