Let f(x)=ln(x). Prove that ∀ x ∈ (1,∞),
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Problems 5) Let (X, M, u) be a measure space, and f e Lt. Assume that S fdu = 1. Prove that 00, 0<a<1, lim n ln (1 +(${2))a) du(x) = { 1, a = 1, 10. a 1. Hint: Use Fatou's lemma for a < 1 and LDCT for a > 1 (dominate by af). 1+00)
1. Let x, a € R. Prove that if a <a, then -a < x <a.
Let f and g be differentiable on R such that f(1) = g(1), and f'(x) < '() for all r ER. Prove that f(x) = g(2) for 3 >1.
Problem 10(20). Let x and y be vectors in R". Prove that |x"y| < ||x|||y- No work, no credit, messy work, no credit, missed steps, no credit disorganized work, no credit.
5-1. Let U ~ Uniform(0,1) and X = – ln(1 – U). Show that The CDF of X is Fx(x) = 1 – e-X, 0 < x < 0 In other word, X is exponentially distributed with 2 = 1.
Let f(x) = cxe-x if x 20 and f(x) = 0 if x < 0. (a) For what value of c is fa probability density function? (b) For that value of c, find P(1<x< 4). 0.368
2. (10 marks) Prove that f(x) = 6 ln(x – 11) is not uniformly continuous on (0,00)
2. (10 marks) Prove that f(x) = 5 ln(x – 7) is not uniformly continuous on (0,00).
Let X be a continuous random variable. Prove that: P(21-; < X < xạ) = 1 - a.
1 xe Let f(x)={? x 8. Prove that f(x) continuous only at +1. Let f(x)= $3.x xs! x >1 Using the definition prove lim f(x)=1 and lim f (x) = 3 x>17 11°