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Let Ich be a not cmety interval and We will say that is a Darboux function ie for any ab,a,beI and any between la) a...
Problem 2 Let f be an absolutely continous function on (0, 1], and f E LP on 0,. Show that, for s...
Please write clearly
Problem 2 Let f be an absolutely continous function on (0, 1], and f E LP on 0,. Show that, for sotne a >D and C>0, we have for any z.y e 0, 1 that
Problem 2 Let f be an absolutely continous function on (0, 1], and f E LP on 0,. Show that, for sotne a >D and C>0, we have for any z.y e 0, 1 that
(3) Let the function f(x) 0 if x Z, but for n e z we have f(n) . Prove that for any interval [a3]...
Advanced Calculus
(3) Let the function f(x) 0 if x Z, but for n e z we have f(n) . Prove that for any interval [a3] the function f is integrable and Ja far-б. Hint: let k be the number of integers in the interval. You can either induct on k or prove integrability directly from the definition or the box-sum criterion.
(3) Let the function f(x) 0 if x Z, but for n e z we have f(n) ....
a) Show that [a,b] | ab. b) Let d be a common divisor of a and...
a) Show that [a,b] | ab.
b) Let d be a common divisor of a and b. Show that
.
c) Prove that (a,b)*[a,b] = ab.
d) Prove that if c is a common multiple of a and b, then
such that k[a,b] = c.
e) Suppose that c is a common multiple of a and b. Show that ab
| (a,b)*c
Defn: Let m e Z. We say that m is a common multiple of a and b if...
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 C>0, then 7 is also integrable on la,b] (6 Marks)...
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 C>0, then 7 is also integrable on la,b] (6 Marks) (2) If C 0, i, still integrable (assuming f(x)关0 for any x E [aM)? If yes, supply a short proof. If no, give a counterexample. (6 Marks)
12. Let f be integrable on a closed interval [a, b]....
Question 4* (Similar to 18.1) Suppose f is a continuous function on a closed interval [a,...
Question 4* (Similar to 18.1) Suppose f is a continuous function on a closed interval [a, b]. In class, we proved that f attains its maximum on that interval, i.e. there exists Imar E la, so that f(Imar) > f(x) for all r E (a,b]. We didn't prove that f attains its minimum on the interval, but I claimed that the proof is similar. In fact, you can use the fact that f attains its maximum on any closed interval...
show steps, thanks » Additional Problem 4. We say that mp is the pth quantile of...
show steps, thanks
» Additional Problem 4. We say that mp is the pth quantile of the distribution function F if Find m, for the distribution having the following density functions: (a) f(z) = 5e-5e, X 〉 0 (b) f(z) = 3, 0 〈 x 〈 2. (c) f(x) =ー2ー r+1 ,一1 < x 〈 1. Answers: (a) -r In (1-p), (b) 2p1/4, (c)-1 +2, P » Additional Problem 5. Suppose that X is equally likely to take any of...
Let k 21 be a positive integer, and let r R be a non-zero real number. For any real number e, we ...
Let k 21 be a positive integer, and let r R be a non-zero real number. For any real number e, we would like to show that for all 0 SjSk-, the function satisfies the advancement operator equation (A -r)f0 (a) Show that this is true whenever J-0. You can use the fact that f(n) = crn satisfies (A-r)f = 0. (b) Suppose fm n) satisfies the equation when m s k-2 for every choice of c. Show that )...
(Limit of functions) Let f : 2-» C be a function, and assume that D(a, r)...
(Limit of functions) Let f : 2-» C be a function, and assume that D(a, r) C Q. We say that lim f(z) L Ď(a, 6) we have |f(z) Ll < e. if for any e > 0 there exists 6 > 0, such that for any z e (a) State the negation of the assertion "lim^-,a f(z) = L". (b) Show that lim- f(z) L if and only if for any sequence zn -» a, with zn a for...
Please all thank you Exercise 25: Let f 0,R be defined by f(x)-1/n, m, with m,nENand n is the minimal n such that m/n a) Show that L(f, P)0 for all partitions P of [0, 1] b) Let mE N. Show that t...
Please all thank you
Exercise 25: Let f 0,R be defined by f(x)-1/n, m, with m,nENand n is the minimal n such that m/n a) Show that L(f, P)0 for all partitions P of [0, 1] b) Let mE N. Show that the cardinality of the set A bounded by m(m1)/2. e [0, 1]: f(x) > 1/m) is c) Given m E N construct a partition P such that U(f, Pm)2/m. d) Show that f is integrable and compute Jo...
Consider the sample space Ω-10, 1]. Let P be a probability function such that for any...
Consider the sample space Ω-10, 1]. Let P be a probability function such that for any interval [a, b], P([a, b) b- a. In other words, probabilty of any interval is its length. Let us start with Co 10, 1], and at nth step, we define C, by removing an interval of length 1/3° from the middle of each interval in Cn-1. For example, G = [0, 1/3ju [2/3, 11, c2 [0, 1/9] U [2/9, 1/3] U [2/3,7/9] U [8/9,...
Please write clearly
Problem 2 Let f be an absolutely continous function on (0, 1], and f E LP on 0,. Show that, for sotne a >D and C>0, we have for any z.y e 0, 1 that
Problem 2 Let f be an absolutely continous function on (0, 1], and f E LP on 0,. Show that, for sotne a >D and C>0, we have for any z.y e 0, 1 that
Advanced Calculus
(3) Let the function f(x) 0 if x Z, but for n e z we have f(n) . Prove that for any interval [a3] the function f is integrable and Ja far-б. Hint: let k be the number of integers in the interval. You can either induct on k or prove integrability directly from the definition or the box-sum criterion.
(3) Let the function f(x) 0 if x Z, but for n e z we have f(n) ....
a) Show that [a,b] | ab.
b) Let d be a common divisor of a and b. Show that
.
c) Prove that (a,b)*[a,b] = ab.
d) Prove that if c is a common multiple of a and b, then
such that k[a,b] = c.
e) Suppose that c is a common multiple of a and b. Show that ab
| (a,b)*c
Defn: Let m e Z. We say that m is a common multiple of a and b if...
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 C>0, then 7 is also integrable on la,b] (6 Marks) (2) If C 0, i, still integrable (assuming f(x)关0 for any x E [aM)? If yes, supply a short proof. If no, give a counterexample. (6 Marks)
12. Let f be integrable on a closed interval [a, b]....
Question 4* (Similar to 18.1) Suppose f is a continuous function on a closed interval [a, b]. In class, we proved that f attains its maximum on that interval, i.e. there exists Imar E la, so that f(Imar) > f(x) for all r E (a,b]. We didn't prove that f attains its minimum on the interval, but I claimed that the proof is similar. In fact, you can use the fact that f attains its maximum on any closed interval...
show steps, thanks
» Additional Problem 4. We say that mp is the pth quantile of the distribution function F if Find m, for the distribution having the following density functions: (a) f(z) = 5e-5e, X 〉 0 (b) f(z) = 3, 0 〈 x 〈 2. (c) f(x) =ー2ー r+1 ,一1 < x 〈 1. Answers: (a) -r In (1-p), (b) 2p1/4, (c)-1 +2, P » Additional Problem 5. Suppose that X is equally likely to take any of...
Let k 21 be a positive integer, and let r R be a non-zero real number. For any real number e, we would like to show that for all 0 SjSk-, the function satisfies the advancement operator equation (A -r)f0 (a) Show that this is true whenever J-0. You can use the fact that f(n) = crn satisfies (A-r)f = 0. (b) Suppose fm n) satisfies the equation when m s k-2 for every choice of c. Show that )...
(Limit of functions) Let f : 2-» C be a function, and assume that D(a, r) C Q. We say that lim f(z) L Ď(a, 6) we have |f(z) Ll < e. if for any e > 0 there exists 6 > 0, such that for any z e (a) State the negation of the assertion "lim^-,a f(z) = L". (b) Show that lim- f(z) L if and only if for any sequence zn -» a, with zn a for...
Please all thank you
Exercise 25: Let f 0,R be defined by f(x)-1/n, m, with m,nENand n is the minimal n such that m/n a) Show that L(f, P)0 for all partitions P of [0, 1] b) Let mE N. Show that the cardinality of the set A bounded by m(m1)/2. e [0, 1]: f(x) > 1/m) is c) Given m E N construct a partition P such that U(f, Pm)2/m. d) Show that f is integrable and compute Jo...
Consider the sample space Ω-10, 1]. Let P be a probability function such that for any interval [a, b], P([a, b) b- a. In other words, probabilty of any interval is its length. Let us start with Co 10, 1], and at nth step, we define C, by removing an interval of length 1/3° from the middle of each interval in Cn-1. For example, G = [0, 1/3ju [2/3, 11, c2 [0, 1/9] U [2/9, 1/3] U [2/3,7/9] U [8/9,...
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