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# HW4 - IE 111 Fall Semester 2007 Homework#4 Due Monday 10/1...

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IE 111 Fall Semester 2007 Homework #4 Due Monday, 10/1 at start of class Question 1. Suppose we have a biased coin such that P(head) P(tail). Suppose I flip the coin three times. Let the random variable X be the number of heads in the 3 flips. You are given the following information: P(X=0) = 2/27 P(X=1) = 6/27 P(X=2) = ? P(X=3) = 9/27 a) Find P(X=2) b) Find P(X 1) c) Find P( 1 X < 2) d) Find P( X=1 | X 1) Question 2. A die has 4 sides (not 6 like a regular die). The four sides are labeled 1, 3, 5, and 10 respectively. It is equally likely that you will get a 1 or a 3. It is equally likely that you will get a 5 or a 10. It is three times more likely that you will get a 5 or10 than a 1 or 3. a) Let X = the outcome of a roll of the die. Find the probability mass function of X. b) If Y=2X 2 +1, find P Y (y) c) If I roll the die 10 times, what is the probability I get exactly 3 tens? Question 3. Let the random variable X be the value of the up face of a certain unfair die. For this die, the probability of getting a specific number (1 through 6) is proportional to twice that number. Thus getting a “5” is 10 times as likely as getting a “1”, the probability of a “4” is four times the probability of a “2”, etc. a) Find the Probability Mass Function P X (x). b) Find P(X 5) c) P(X 4 | 3 < X 5) d) Find the probability of rolling a number greater than or equal to “4” three times in a row.

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