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Homework Solutions 04

# Homework Solutions 04 - (W4150 Intro to Probability and...

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Solutions to Homework #4 (W4150 ‘Intro to Probability and Statistics’, S04) Sec. 4.1. (1) In this problem we have to integrate joint distribution density function multiplied by x with respect to both x and y - but one should be very careful when setting limits for integration. You may choose any of these variables to be ’free’ in the sense that it can go from –a to a , but the second variable must be restricted with respect to the inequality x 2 + y 2 < a 2 . So if you take, for example, y as a free variable, you’ll end up with E[X] = 0 2 2 1 1 2 2 2 2 2 2 2 2 2 2 = ° ± ² ³ ´ µ · ¸ ¹ ¹ º » - - · ¸ ¹ ¹ º » - = ¼ ¼ ¼ - - - - - dy y a y a a xdxdy a a a y a y a a a π π Sec.4.1. (4) Assigning weights of 3 w and w for a head and tail, respectively, we obtain P(H) = ¾ and P(T) = ¼. The sample space for the experiment is S = {HH, TH, HT, TT}. Now if X represents the number of tails that occur in two tosses of the coin, we have P(X=0) = P(HH) = (3/4)(3/4) = 9/16; P(X=1) = P(HT) + P(TH) = 2(3/4)(1/4) = 3/8; P(X=2) = P(TT) = (1/4)(1/4) = 1/16.

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