6. Show that for any sequence of events (F)y-1
Show that for any sequence of events (F)j-1, TL TL 8 & i-1
6. Show that if A1, A2, ... is an expanding sequence of events, that is, AC A₂C...... then P(ALU AQU....) = lim P(An). 1-00
For a probability space (Ω,F, P. ifB. Be' . . . is a sequence of events such that Ση i P(Bk) 〉 n ї. show that Pîne i Bk) 〉 0
a) Show that f is discontinuous at any x 6= 0. b) Show that f is continuous at x = 0. c) Show that f is differentiable at x = 0 and compute the value f 0 (0). d) Show that f is not integrable on the interval [1, 2] (or any interval, but I don’t mind if you use that interval specifically). (x2 (x EQ) f(x)=o (x &Q)
For any sequence of RVs {Xn}. Show that max|Xk|→0 in pr →n^-1Sn→0 in pr.
please prove Let - andd On -n+1 Show that a is an increasing sequence, that bn is a decreas- Let - andd On -n+1 Show that a is an increasing sequence, that bn is a decreas-
1 7) Show that the series converges/diverges 1 1 8) Show that the sequence a N + 1 N+ 1 is monotonic.
(3) Let (2,A, /i) be a measure space. Let f : N > R* be a nonnegative measurable function. Define the sequence fn(x) = min{f(x), n}, n E N. Prove that for any A E A f du lim fn du A 4 (You must show that the integrals exist.) (3) Let (2,A, /i) be a measure space. Let f : N > R* be a nonnegative measurable function. Define the sequence fn(x) = min{f(x), n}, n E N. Prove...
Show, by Minkowski diagrams, that (1) time-like events can occur at the same position, but not at the same time, (2) space-like events can occur at the same time, but not at the same position, (3) space-like events can have any order/sequence.