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practice_exam1

# practice_exam1 - 1 List Kolmogorov’s axioms 2 What is an...

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Unformatted text preview: 1. List Kolmogorov’s axioms. 2. What is an event? Give me the general definition, not an example. 3. Suppose we select k objects out of n objects. How many ways can we do this in the following cases? a. Unordered, with replacement. b. Unordered, without replacement c. Ordered, with replacement d. Ordered, without replacement 4. Let Ω = [0 , 1], A = [0 ,. 9], B = ( . 3 ,. 4], C = [ . 35 ,. 5] ∪ [ . 7 ,. 9), D = [0 ,. 8], E = [ . 2 ,. 5], and let P ([ a,b ]) = b- a be a probability measure on Ω. a. Calculate C ∩ D b. Calculate A ∪ B . c. Calculate E ∩ ( A ∪ D ). d. Calculate P ( E | D ). e. Calculate P ( C | E ). 5. Let Ω = (-∞ , ∞ ), A = [0 , 1000), B = (-∞ ,- 100], and C = [- 500 , 500]. Make a Venn diagram of this situation. 6. Let Ω = [0 , 5]. Find a partition of Ω consisting of at least 3 events. 7. Let A and B be independent events. Prove or disprove that A is necessarily independent of B . 8. Let A , B , and C be events such that A is independent of B and A is independent of C . Prove or disprove that A is necessarily independent of B ∪ C . 9. Suppose P ( A ) = . 2, P ( B ) = . 3, and P ( A ∪ B ) = . 44. Are A and B independent? 10. Suppose Factor A has 3 levels, A 1 , A 2 , and A 3 , and Factor B has 3 levels, B 1 , B 2 , and B 3 . Also, suppose P ( A 1 ) = . 2, P ( A 2 ) = . 7, P ( A 3 ) = . 1, P ( B 1 ) = . 3, P ( B 2 ) = . 6, P ( B 3 ) = . 1, and Factor A is...
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practice_exam1 - 1 List Kolmogorov’s axioms 2 What is an...

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