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hw3soln - Homework 3 Solutions The George Washington...

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Homework # 3 Solutions The George Washington University ApSc 116 1 / 10
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Problem (Page 102, # 3) Suppose that two balanced dice are rolled, and let X denote the absolute value of the difference between the two numbers that appear. Determine and sketch the p.f. of X. The r.v. X has sample space S = { 0 , 1 , 2 , 3 , 4 , 5 } . By looking over the 36 possible outcomes enumerated in Example 1.6.4, we find that X = 0 for 6 outcomes, X = 1 for 10 outcomes, X = 2 for 8 outcomes, X = 3 for 6 outcomes, X = 4 for 4 outcomes, and X = 5 for 2 outcomes. Hence, the p.f. f ( x ) is as follows: x 0 1 2 3 4 5 f ( x ) 3/18 5/18 4/18 3/18 2/18 1/18 2 / 10
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Problem (Page 109 # 8) Suppose that the p.d.f. of a random variable X is as follows: f ( x ) = ce - 2 x for x > 0 , 0 otherwise. a. Find the value of the constant c and sketch the p.d.f. b. Find the value of Pr (1 < X < 2) . Solution: a. We must have -∞ f ( x ) dx = 0 cexp ( - 2 x ) dx = 1 2 c = 1 . Therefore, c = 2. This p.d.f. has the appearance in the figure in the next slide. 3 / 10
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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.5 1 1.5 2 2.5 3 3.5 4 2 exp(-2x) x (2 * exp(-2*x)) b. Here we have Pr (1 < X < 2) = 2 1 f ( x ) dx = exp ( - 2) - exp ( - 4) = 0 . 117 4 / 10
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