Midterm - Indian Statistical Institute Delhi Centre Master...

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Unformatted text preview: Indian Statistical Institute, Delhi Centre Master of Statistics (Applications) Measure Theoretic Probability Spring 2009 Mid-Term Examination Instructor: Antar Bandyopadhyay Date: March 4, 2009 Time: 10:00 AM - 12:00 PM Note: • Please write your name and roll number on top of your answer paper. • There are 4 problems carrying a total of 30 points. Solve as many as you can. The maximum you can score is 20. Show all your works and write explanations when needed. • This is an open note examination. You are allowed to use your own hand written notes (such as class notes, your homework solutions, list of theorems, formulas etc). Please note that no printed materials or photo copies are allowed, in particular you are not allowed to use books, photocopied class notes etc. Total Points: 20 Duration: 2 Hours 1. Fix n ≥ 1 and consider the measure space (Rn , BRn , λn ) where λn is the n-dimensional Lebesgue measure. Let D be a n × n diagonal matrix with non-zero diagonal entries (d1 , d2 , . . . , dn ). For any subset A ⊆ Rn deﬁne D (A) := Dx x ∈ A . (a) Show that D (A) ∈ BRn for all A ∈ BRn . n [4] |di | i=1 (b) For any A ∈ BRn show that λn (D (A)) = λn (A). [6] 2. Show that there exists a probability P on (0, 1], B(0,1] such that P ({q }) > 0 for all q ∈ Q ∩ (0, 1], and P (Q ∩ (0, 1]) = 1. [10] 3. Let Ω = ∅ and F 1 , F 2 be two σ -algebras on Ω. Show that σ (F 1 ∪ F 2 ) = σ (U ) where U := A1 ∪ A2 A1 ∈ F 1 and A2 ∈ F 2 . [5] 4. Let Ω = N and F be the ﬁnite-co-ﬁnite algebra on Ω. Let µ : F → [0, 1] be a set function deﬁned as µ (A) = 0 1 if |A| < ∞ . otherwise F = µ. [5] Show that there is no measure ν on (Ω, σ (F )) such that ν G ood Luck 1 ...
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