Homework 3

# Homework 3 - n balls have been dropped Problem 4 The annual...

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FALL 2011 Instructor: Sujay Sanghavi Homework 3 Due: September 22th in class Problem 1 Count the number of distinguishable ways in which you can arrange the letters in the words: (a) children (b) bookkeeper Problem 2 In how many ways can 8 people be seated in a row if (a) there are no restrictions on the seating arrangement; (b) persons A and B must sit next to each other; (c) there are 4 men and 4 women and no 2 men or 2 women can sit next to each other; (c) there are 5 men and they must sit next to each other; (d) there are 4 married couples and each must sit together? Problem 3 Tom has n labeled balls and n labeled bins. For each of the balls, he chooses any one of the bins with equal probability (and independently of any other decision) and drops the ball in the chosen bin. Determine the probability that exactly one of the bins is empty after all
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Unformatted text preview: n balls have been dropped. Problem 4 The annual premium of a special kind of insurance starts at \$1000 and is reduced by 15% after each year where no claim has been ﬁled. The probability that a claim is ﬁled in a given year is 0.1, independently of preceding years. What is the PMF of the total premium paid up to and including the year when the ﬁrst claim is ﬁled? Problem 5 Let X be a discrete random variable that is uniformly distributed over the set of integers in the range [ a,b ] , where a and b are integers with a < < b . Find the PMF of the random variables max { ,X } and min { ,X } . (Note that max { ,X } denotes a random variable that takes the value of X if this value is bigger than 0, and 0 otherwise. Similarly for min { ,X } .)...
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