Lecture1_answer

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Lecture 1 notes answer

Question 4: Answer ◼ 4.1: If a coin is flipped once, what is the size of the sample space? A ◼ 4.2: If a coin is flipped three times, what is the size of the sample space? D ◼ {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} ◼ 4.3 If a dice is tossed once, what is the size of the sample space? C ◼ A. 2 B. 4 C. 6 D. 8

Question 5 ◼ k=3 #no of coin flips ◼ n=1000 # repeat the experiment n times ◼ How can we find the histogram? ◼ Hint: ◼ Consider k=2 first, what is the size of sample space?

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K=3: 8 possible cases ◼ 1st toss: H → 1, 2nd toss: H → 1, 3 rd : H:1 ◼ Sum = 1 + 1 +1= 3 ◼ 1st toss: H → 1, 2nd toss: H → 1, 3 rd : T:-1 ◼ Sum = 1 + 1 + -1= 1 ◼ 1st toss: H → 1, 2nd toss: T → -1, 3 rd : H:1 ◼ Sum = 1 + -1 +1= 1 ◼ 1st toss: H → 1, 2nd toss: T → -1, 3 rd : T:-1 ◼ Sum = 1 + -1 + -1= -1

K=3: 8 possible cases ◼ 1st toss: T → -1, 2nd toss: H → 1, 3 rd : H:1 ◼ Sum = -1 + 1 +1= 1 ◼ 1st toss: T → -1, 2nd toss: H → 1, 3 rd : T:-1 ◼ Sum = -1 + 1 + -1= -1 ◼ 1st toss: T → -1, 2nd toss: T → -1, 3 rd : H:1 ◼ Sum = -1 + -1 +1= -1 ◼ 1st toss: T → -1, 2nd toss: T → -1, 3 rd : T:-1 ◼ Sum = -1 + -1 + -1= -3

Question 5: answer ◼ K=1, X=1 (head), X=-1 (Tail) ◼ K=3 ◼ {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} ◼ → {3, 1, 1, -1, 1, -1, -1, -3} ◼ P(X=3) = P(X=-3) = 1/8, P(X=1) = P(X=-1)=3/8 ◼ P(HHH) = P(H)P(H)P(H) = 1/8 -3 -1 1 3

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6.1: Answer ◼ Z = X + Y. ◼ Determine the probability density function of Z. Y\X 1 2 3 4 1 2 3 4 5 2 3 4 5 6 3 4 5 6 7 4 5 6 7 8 Z P(Z) 2 1/16 3 2/16 4 3/16 5 4/16 6 3/16 7 2/16 8 1/16

6.2: Answer ◼ M = max(X, Y) ◼ Min(X, Y) = 2 Y\X 1 2 3 4 1 2 3 4 5 2 3 4 5 6 3 4 5 6 7 4 5 6 7 8 If we know that min(X,Y) = 2, what is The prob that M=1? Prob = 0 The prob that M=3? Prob = 2/5 ◼ P(A|B) = P(A and B) / P(B) ◼ P(M=3 | B) = P(M=3 and B) / P(B) ◼ P(M=3 and B) =2/16, P(B) = 5/16

Question 7.1: Answer ◼ Are X and Y independent? y=-1 y=0 y=1 x=-1 0.4 0.16 0.24 x=0 0.05 0.02 0.03 x=1 0.05 0.02 0.03 X -1 0.8 0 0.1 1 0.1 Y -1 0.5 0 0.2 1 0.3 y=-1 y=0 y=1 x=-1 0.8x0.5=0.4 0.8x0.2=0.16 0.8x0.3=0.24 x=0 0.1x0.5=0.05 0.1x0.2=.02 0.1x0.3=0.03 x=1 0.1x0.5=0.05 0.1x0.2=0.02 0.1x0.3=0.03 ◼ P(x and y) = P(x) P(y)

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Lecture1_answer

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