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By Klotz J.H.

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We have N Ai ) = P ( P( i=1 ∞ Ai ) = i=1 ∞ N P (Ai ) P (Ai ) = i=1 i=1 and finite additivity holds for disjoint events. 3 P (Ac ) = 1 − P (A). Proof. Using S = A ∪ Ac , where A ∩ Ac = φ, and finite additivity 1 = P (S) = P (A ∪ Ac ) = P (A) + P (Ac ) . Subtracting P (A) from both sides gives the result. 4 If A ⊂ B, then A = A ∩ B and P (A) ≤ P (B). 3. PROBABILITY Proof. e ∈ A =⇒ e ∈ A and e ∈ B =⇒ e ∈ A ∩ B . Converseley e ∈ A ∩ B =⇒ e ∈ A and e ∈ B =⇒ e ∈ A so that A = A ∩ B. Next using the distributive law for events P (B) = P (B ∩ S) = P (B ∩ (A ∪ Ac )) = P ((B ∩ A) ∪ (B ∩ Ac )) and using A = A ∩ B and finite additivity for disjoint A and B ∩ Ac = P (A ∪ (B ∩ Ac )) = P (A) + P (B ∩ Ac ) ≥ P (A) .

4). 13. 1. 4 4 6 10 12 5 14. Calculate the sample median. 15. Calculate the Walsh sum median. 16. Calculate the sample mean. 17. 13). 18. 4). 19. 13). 20. Prove D ≤ R/2. 1 The Sample Space Consider a random experiment which has a variety of possible outcomes. Let us denote the set of possible outcomes, called the sample space, by S = {e1 , e2 , e3 , . }. If the outcome ei belongs to the set S we write ei ∈ S. If the outcome does not belong to the set S we write e ∈ S. Ws say the sample space is discrete if either there is a finite number of possible outcomes S = {ei : i = 1, 2, .

19. 13). 20. Prove D ≤ R/2. 1 The Sample Space Consider a random experiment which has a variety of possible outcomes. Let us denote the set of possible outcomes, called the sample space, by S = {e1 , e2 , e3 , . }. If the outcome ei belongs to the set S we write ei ∈ S. If the outcome does not belong to the set S we write e ∈ S. Ws say the sample space is discrete if either there is a finite number of possible outcomes S = {ei : i = 1, 2, . . , n} or there is a countably infinite number of possible outcomes (the outcomes can be put into one to one correspondence with the set of positive integers) S = {ei : i = 1, 2, .

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