Problem Sheet 1 Probability Theory and Mathematical Statistics (2017) 1. Solid Foundations Problem 1. Let Ω = [0; 1] and let F be a σ-algebra such that the half-open intervals An = [0; 1=n) 2 F for all n ≥ 1. Show that (i) The singleton f1g 2 F (ii) The half-open interval [1=4; 1=2) 2 F (iii) f0g 2 F (iv) Is it true that f1=3g 2 F? Problem 2. Let A and B be disjoint subsets of Ω such that C = (A [ B)c 6= ;. Describe the σ-algebra generated by A and B. How many elements does it contain? Problem 3. Let A and B be two events, with P(A) = 0. Show that (i) P(A [ B) = P(B) (ii) P(Ac \ B) = P(B) Problem 4. Let Ω be a set, A; B ⊂ Ω its subsets. Suppose that D = A \ B 6= ; and C = (A [ B)c 6= ;. Describe the σ-algebra F generated by the sets A and B and list all the elements of F. How many elements are there in F? Problem 5. Suppose we have the following three models for the cast of a fair die: (i) Ω = f1; 2; 3; 4; 5; 6g, where the elementary event i 2 Ω represents the outcome `the number on the die is i', for i = 1;:::; 6. (ii) Ω = fa; b; cg, where the elementary event a represents the outcome `the number on the die is 1 or 2', elementary event b, `... 3 or 4', and elementary event c, `... 5 or 6'. (iii) Ω = f!1;!2;!3g, where the elementary event !1 represents the outcome `the number on the die is 1,2 or 3', elementary event !2, `... 4 or a 5', and elementary event !3, `... 6'. In each case, assign a probability to Ω so that the model corresponds to our random experiment. For each of the three models, say if A, B, C and D are events, and, when it is so, express them as a subset of Ω and give their probability: A = fthe number on the die is less or equal to 4g B = fthe number is a 6g C = fthe number on the die is a 1 or a 2g D = feven numberg: 1 Problem Sheet 1 Probability Theory and Mathematical Statistics (2017) Problem 6. Suppose that for events A and B, we know the following probabilities P(A) = 0:4 P(B) = 0:5 P(A [ B) = 0:7 : Compute (i) P(Ac) (ii) P(Ac \Bc) (iii) P(A\B) (iv) P(Ac j B) (v) P(A j Bc) Problem 7. Prove or disprove (by giving a counterexample) the following assertions: (i) If P(A) = P(B) = 1, then P(A∆B) = 0, where A∆B := (A \ Bc) [ (Ac \ B). (ii) If P(A n B) = P(B n A) then P(A) = P(B), where A n B := A \ Bc. (iii) If P(A n B) = 2=3 and P(B n A) = 2=3 then we are in trouble. Problem 8. Let B, C and D be possibly overlapping events, and define a sequence fAngn≥1 of events as 8 B n = 1; 2; 3 <> 2 An = C n = k for k ≥ 2 :>D for all other values of n : Express in terms of B; C and D the event fAn i:o:g. Problem 9. Suppose we have three cards: one is red on both sides, one is white on both sides, and one has one side red and one side white. Draw one card randomly and put it one the table, one face up. The face showing is red. Would you bet that the hidden face is white? Hint: Describe the sample space Ω precisely, then compute the required conditional probability. Problem 10. Prove that there does not exist a uniform distribution on R. Uniformity means that for any event A, P(A) = P(A + x), for any x 2 R, where A + x = fa + x j a 2 Ag. Problem 11. Let Ω = R, F = B(R) and P be the distribution with density f(x) = e−x ; x > 0 : We definite X(!) := b!c be the integer part of !, that is X(!) = k if k ≤ ! < k + 1, k 2 Z. (i) Show that X is a random variable on (R; B(R)). (ii) Describe σ(X), the σ-algebra generated by X. (iii) Find the distribution PX of X. Problem 12. Let X1;:::;Xn be a random sample of i.i.d. random variables, where each Xj has distribution function F . Denote by X(1) ≤ X(2) ≤ ::: ≤ X(n) the order statistics of the sample. (i) Find the distribution function of X(1). 2 Problem Sheet 1 Probability Theory and Mathematical Statistics (2017) (ii) Find the distribution function of X(n). (iii) Find the joint distribution function of (X(1);X(n)). (iv) Assuming F has density f, derive the density of X(k), for 1 ≤ k ≤ n. Hint: Express the event fX(k) 2 (x; x + ∆]g for ∆ > 0 in terms of the Xjs, and use this representation to find the probability of the event. Then divide by ∆ and take the limit as ∆ ! 0. Problem 13. Let X = (X1;X2) be a random vector with uniform distribution over the domain 2 2 f(x1; x2) j x1 + x2 ≤ 1g. Consider the events A1 = fX1 > 0g, A2 = fX2 > 0g and A3 = fX1X2 > 0g. Show that these events are pairwise independent, but not jointly independent. 3.