Let 2 N (1,2,3,...} be a sample space and F-2N a sigma algebra. . . ....
Let F be a o-algebra of subsets of the sample space S2. a. Show that if Ai, A2, E F then 1A, F. (Hint use exercise 4) b. Let P be a probability measure defined on (2, F). Show that
Let C,C Є F where F is a sigma algebra on Ω with a probability measure P. Show that F1={ⱷ, Ω ,C,Cc} and F2={ ⱷ, Ω ,D,Dc } are independent iff C and D are independent?
4. Consider the sample space S 1,2,3,...), and assume that outcomes have the probabilities P(i)- 2-'. For any n 2 0, define the discrete random variable Xn S0,... , n) by x,(i)-1 mod (n + 1), where mod means"modulo (a) Show that Xn converges in probability to the "identity" random variable X, defined by X(i)-. (b) Show that Xn converges in distribution to the Geom (1/2) random variable (e.g. to the time of the first Head in a sequence of...
5. Consider an experiment in which the sample space Ω is precisely the real line 9t -(-00,00). Let B denote the Borel σ-algebra on the sample space Ω, ie., B is the σ-algebra generated by all the open intervals of the form (a,b), for-oo 〈 a 〈 b〈00, (a) Show that 3 contains all closed intervals of the form lp,91 for all-oo 〈 pくqく00, (b) Show that B contains all finite collections {xi,x2, - .. ,xn) of n distinct real...
Consider the sample space Ω-1. 2. 3. 1. Let P((1) , P(2) and P({3)-a. a) Are there any values of a for which the event A 1,21 is dependent of B-3,4? In such a case find those values b) Are there any values of a for which the event A ,2] is independent of C 2,3? In such a case find those values. c Are there any values of a for which the event C 2,3 is dependent of B-3,4?...
5. Let Ω = { 1, 2, 3, . j be the countably infinite sample space whose elements (outcomes) are the positive integers. For each positive integer n, define the event An k k is a multiple of n \ (a) Find n and m such that An-Ag n A4 and Am-A6 A9. (b) If P((k]) -fnd the probability of the event As k-1 Note: an exact answer is required here; if you write a program to obtain answers of...
Let V = R3[x] be the vector space of all polynomials with real coefficients and degress not exceeding 3. Let V-R3r] be the vector space of all polynomials with real coefficients and degress not exceeding 3. For 0Sn 3, define the maps dn p(x)HP(x) do where we adopt the convention thatp(x). Also define f V -V to be the linear map dro (a) Show that for O S n 3, T, is in the dual space V (b) LetTOs Show...
Problem 4 (20p). Let α > 0, and for each n E N let Xn : Ω R b probability space (2, F, P) with the gamma distribution Ta,n. Does there exist a random variable e a random variable on a Problem 4 (20p). Let α > 0, and for each n E N let Xn : Ω R b probability space (2, F, P) with the gamma distribution Ta,n. Does there exist a random variable e a random variable...
8 arbitrary set. K is Cousider E} n=1 nieU and Let (X, K) be a measure space where X is an sigma-algebra of subsets of X and is a measure sequenc o clemenis of K We delin lim supn(Fn) liminfn(En)- U then prove: (a) lim in(E)) lim inf(u(E,) (b) T J (c) If sum E,)x, then (lim sup(E)) = 0 x X) <oc lor somc nE N, then lim supn (Fn)> lim sup(u(F,n )) 8 arbitrary set. K is Cousider...