d X n oo Show lim sup,P(X>k)0 when X Here cdf of X, and X are continuous for every real number. d X n oo Show l...
0, oo) which converges to a certain real Let f be a real-valued continuous function over o0, i.e., lim f(x) = A. Answer the following questions value A as Find the following limit lim aoo a2 f (x)dx 0, oo) which converges to a certain real Let f be a real-valued continuous function over o0, i.e., lim f(x) = A. Answer the following questions value A as Find the following limit lim aoo a2 f (x)dx
where Problem 36. Assume f : X → [0, oo]. Prove that if Σ f(x) < 00, then {x E X (z) > 0} is a countable set. (HINT: Show that for every k E N the set {x E X | f(x) > k-1} is finite.) f(x)-sup f(x) | F is any finite subset of X TEF Problem 36. Assume f : X → [0, oo]. Prove that if Σ f(x) 0} is a countable set. (HINT: Show that...
Problem 5. Suppose that the continuous random variable X has the distribution fx(z),-oo < x < oo, which is symmetric about the value x-0. Evaluate the integral: Fx (t)dt -k where Fx(t) is the CDF for X, and k is a non-negative real number.
Problem 5. Suppose that the continuous random variable X has the distribution fx(x), -00 <oo, which is symmetric about the value r 0. Evaluate the integral: Fx (t)dt -k where Fx(t) is the CDF for X, and k is a non-negative real number. Hint: Use integration by parts
Question 1 1. [5 pts] Give a complete definition of lim f(x) = -oo if... 2. [25 pts] Give an example of each of the following, or state one or more theorems which show that such an example is impossible: a. A countable collection of nonempty closed proper subsets of R whose union is open. b. A nonempty bounded subset of R with no cluster points. c. A convergent sequence with two convergent subsequences with distinct limits. d. A function...
Suppose fon (0,1) is uniformly continuous. Show that there is a real number A such that the function F defined by F(O)=A, F(x)=f(x) if x € (0,1), is continuous on (0,1]. (Suggestion: Show first that if {Xn}, Xn € (0,1] has lim xn = 0, then {f(xn)} is a Cauchy no sequence. Then show this sequence has the same limit no matter which {Xn} sequence going to you choose).
1. Find Fx in terms of φ (t). Is X a continuous random variable ? 2. Compute p(X 0) 3. Compute E(X). Hint: use the CDF expectation formula, and integration by parts. You may assume that lim, t"o(-t) 0 for all n 2 0. 4. Find the CDF Fx (u) 5. Compute V(X). Hint: use Fxa, and follow the same hint of part (3) 1. Find Fx in terms of φ (t). Is X a continuous random variable ? 2....
2.13.4 2.13.5 Show that lim supno (-X) = -(liminf ,-Xn). If two sequences {an) and {bn} satisfy the inequality an <b, for all sufficiently large n, show that limsupan Slim sup bn and liminfa, <liminf bn. 100 2.13.6 Show that lim, 100 Xn = o if and only if lim sup.Xn = liminf xn = c. n-00 2.13.7 Show that if lim sup a n = L for a finite real number L and € > 0, then an >...
#s 2, 3, 6 2. Let (En)acy be a sequence in R (a) Show that xn → oo if and only if-An →-oo. (b) If xn > 0 for all n in N, show that linnAn = 0 if and only if lim-= oo. 3. Let ()nEN be a sequence in R. (a) If x <0 for all n in N, show that - -oo if and only if xl 0o. (b) Show, by example, that if kal → oo,...