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# HW4_11 - IE 111 Fall Semester 2011 Homework#4 Due Wednesday...

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IE 111 Fall Semester 2011 Homework #4 Due Wednesday 10/5 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) = 0.343 P(X=1) = 0.441 P(X=2) = ? P(X=3) = 0.027 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 five sides are labeled 1, 3, 5, 10, and 20 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. It is two times as likely that you will get a 20 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. a) Find the Probability Mass Function P X (x). b) Find P(X 2) c) P(X 4 | 3 < X 5) d) Find the probability of rolling a number greater than or equal to “3” four times in a row. e) The Cumulative Distribution Function F X (x) is Defined as F X (x) = P(X x) Find the C.D.F. for the random variable X.

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Question 4. The cumulative distribution function F X (x) of a random variable X is given by: F X (x) = 0 for x < 5 F X (x) = 0.6 for 5 x < 6 F X (x) = 0.8 for 6 x < 10 F X (x) = 1.0 for x 10 a) Find the probability mass function P X (x).
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