Prove the following: where B JA, and X is an integrable random variable
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(4) Use the Box-sum criterion to prove that if f is integrable on [a, b] and is also integrable on |b,e, then f is integrable on la, e) and Je fdr- o fdz+ (5) Suppose that (r) 2 0 and is continuous on a, b). Prove that if f - 0, then f(x) = 0 for all x E a,b]. Hint: Assume to the contrary that there is some r E [a, b] where f(x) > 0. What can...
Three questions!please!
7. Prove that J(x) is integrable on (0,b), and calculate their integral. 8. Prove that the following function is integrable on [0, 1], and calculate the integral. 1 if for some n E N 0 (z)= otherwise. 8. Prove that if f is integrable on (a, b, then f2 is also integrable on la,b
Problem 1.30. Prove the following two properties: 1. If X is integrable and A-measurable, then ElXA X. 2. If X, Y are integrable and a, b E R, then E[aX + bY|A] = aEMA +
Let {h} be a sequence ofRiemann integrable functions on [a,b], such that for each x, {h(x)) is a decreasing sequence. Suppose n) converges pointwise to a Riemann integrable function f Prove that f(x)dxf(x)dx. lim n00
Let {h} be a sequence ofRiemann integrable functions on [a,b], such that for each x, {h(x)) is a decreasing sequence. Suppose n) converges pointwise to a Riemann integrable function f Prove that f(x)dxf(x)dx. lim n00
Problem 5. Prove the following result for any number a and discrete random variable X. 티(X-a 21 = Var(X) + (E(X)-a)2 You must start your proof by using the definition of the expected value of a function of a discrete random variable, i.e. where g(x)- (x-a)
9. Let (2, F,P) be a probability space, X be an square-integrable random variable defined on this space and let G be a sub-g-field of F. Relying only on the definition of conditional expectation, show the following properties: a) E(E(X|9)) = E(X). b) If X is independent of G, then E(X\G) = E(X) a.s.
12. Let f be integrable on a closed interval [a, b]. Suppose that there is a real number C such that f(x) 2C for all E a, b (1) Prove that if С > 0, then, is also integrable on [a,b, (6 Marks) (2) If C 0, i, still integrable (assuming f(x) 0 for any x E [aA)? If yes, supply a short proof. If no, give a counterexample. (6 Marks)
12. Let f be integrable on a closed interval...
31 (a) If fis integrable, prove that fa is integrable. Hint: Given e>0, let h and k be step functions such that h f k and j (k-h) < ε/M, where M is the maximum value of Ik(x) +h(x)]. Then prove that h and k2 are step functions with h' srsk (we may assume that OShSSk since f is integrable if and only if I is-why?), and that I (k2 - h2) <e. Then apply Theorem 3.3. (b) If fand...
Prove that if ? is integrable on [?, ?] and ?(?) ≥ 0 for all ?
in [?, ?], then
[ f(x)dx > 0 7. Prove that if f is integrable on [a, b] and f(x) > 0 for all x in [a, b], then sof(x)dx > 0.
(c) Let f :la,b- R be an integrable function. Prove that lim . (Your argument should include why faf makes sense for a < x < b.)
(c) Let f :la,b- R be an integrable function. Prove that lim . (Your argument should include why faf makes sense for a