Homework Answers
TOPIC: Set theory and Probability.
1. Consider a statistical experiment E: (, F,P) and an event A . Note: A EF....
1. Consider a statistical experiment E: (, F,P) and an event A . Note: A EF. a. Use the axioms of probability to show that P(A) 1-P(A). b. Repeat (a) using the definition of the σ-field. 2. Consider a statistical experiment E: (, F,P) in which a fair coin is flipped successively until the same face is observed on successive flips. Let A = {x: x = 3, 4, 5, . . .); that is, A is the event that...
The % difference = Standard E° (using table) = (show calculation) Ee=Fe+2/ Fe= -0.44V Eo=Cu2+/ Cu=...
The % difference = Standard E° (using table) = (show calculation) Ee=Fe+2/ Fe= -0.44V Eo=Cu2+/ Cu= +0.34 V Eo cell=Eocathode-Eganode = Ee Cu+2/cu- Eo Fe2+/ Fe = +0.34+ 0.44 V = +0.78 V Note that the list of reduction reactions includes the reduction of ironfil) to elemental iron, and also the reduction of iron(III) to iron(II). Is your data more consistent with the formation of Fe? or Felt? Explain your answer.
6. For p > 0, let fe(r) = r-ı (logr)-p. (a) Give p > 0 and e> 0, show that x-1-e 〈 fe(x) < x-1 f...
6. For p > 0, let fe(r) = r-ı (logr)-p. (a) Give p > 0 and e> 0, show that x-1-e 〈 fe(x) < x-1 forsufficiently large (b) For which p does J2cb(x)dx converge.
6. For p > 0, let fe(r) = r-ı (logr)-p. (a) Give p > 0 and e> 0, show that x-1-e 〈 fe(x)
2. Show that P[AIB] satisfies the three axioms of probability b) PISIB] 1 for sample space S c) I...
2. Show that P[AIB] satisfies the three axioms of probability b) PISIB] 1 for sample space S c) If AnC 0 (empty set), then P[An CIB] P[AIB] + P[CIB]
2. Show that P[AIB] satisfies the three axioms of probability b) PISIB] 1 for sample space S c) If AnC 0 (empty set), then P[An CIB] P[AIB] + P[CIB]
a. Define what it means for two logical statements to be equivalent b. If P and Q are two statements, show that the statement ( P) л (PvQ) is equivalent to the statement Q^ P c. Write the converse a...
a. Define what it means for two logical statements to be equivalent b. If P and Q are two statements, show that the statement ( P) л (PvQ) is equivalent to the statement Q^ P c. Write the converse and the contrapositive of the statement "If you earn an A in Math 52, then you understand modular arithmetic and you understand equivalence relations." Which of these d. Write the negation of the following statement in a way that changes the...
Exercise 1. Let f : R R be differentiable on la, b, where a, b R...
Exercise 1. Let f : R R be differentiable on la, b, where a, b R and a < b, and let f be continuous on [a, b]. Show that for every e> 0 there exists a 6 > 0 such that the inequality f(x)- f(c) T-C holds for all c, x E [a, 히 satisfying 0 < |c-x| < δ
If P(E)9 and P(F)-.8, show that P(EnF)2.7. I inequality, namely, n general, prove Bonferroni s Use...
If P(E)9 and P(F)-.8, show that P(EnF)2.7. I inequality, namely, n general, prove Bonferroni s Use induction to generalized Bonferroni's inequality to n events and show the result.
For the given function f(x) and numbers L, c, and e > 0, find an open interval about c on which the inequality fx)-L...
For the given function f(x) and numbers L, c, and e > 0, find an open interval about c on which the inequality fx)-L<e holds. Then give a value for 6 > 0 such that for all x satisfying 0< x -cl <8 the inequality f(x)-LIe holds. f(x) mx, m> 0, L = 2m, c 2, e = 0.03 For what open interval does the inequality f(x)-L hold? <e (Type your answer in interval notation.) Find the largest value 6>0...
Question 3 (Chapter 6) 12+3-3-6 14 marks] Fix p E N and consider the following set: (c) Compute Ci and C2 (d) Show that a 0 is an extreme point of Cp. Hint: you may use (without proof) that the f...
Question 3 (Chapter 6) 12+3-3-6 14 marks] Fix p E N and consider the following set: (c) Compute Ci and C2 (d) Show that a 0 is an extreme point of Cp. Hint: you may use (without proof) that the family of functions fe',e", ..., e) is linearly independent.
Question 3 (Chapter 6) 12+3-3-6 14 marks] Fix p E N and consider the following set: (c) Compute Ci and C2 (d) Show that a 0 is an extreme point of...
3. (25 pts) Let fe C2[a, b], for a < b, and let {p,}0 be Newton's...
3. (25 pts) Let fe C2[a, b], for a < b, and let {p,}0 be Newton's method, where p,n E [a, b] for all n 2 0. Suppose pn Converges top E [a, b], where f(p) 0, f'(p) 0, and p #p for all n 2 0. Find an expression for X 2 0, where sequence generated by a. Pn+1 -p lim = Pn-pl2
3. (25 pts) Let fe C2[a, b], for a
1. Consider a statistical experiment E: (, F,P) and an event A . Note: A EF. a. Use the axioms of probability to show that P(A) 1-P(A). b. Repeat (a) using the definition of the σ-field. 2. Consider a statistical experiment E: (, F,P) in which a fair coin is flipped successively until the same face is observed on successive flips. Let A = {x: x = 3, 4, 5, . . .); that is, A is the event that...
The % difference = Standard E° (using table) = (show calculation) Ee=Fe+2/ Fe= -0.44V Eo=Cu2+/ Cu= +0.34 V Eo cell=Eocathode-Eganode = Ee Cu+2/cu- Eo Fe2+/ Fe = +0.34+ 0.44 V = +0.78 V Note that the list of reduction reactions includes the reduction of ironfil) to elemental iron, and also the reduction of iron(III) to iron(II). Is your data more consistent with the formation of Fe? or Felt? Explain your answer.
6. For p > 0, let fe(r) = r-ı (logr)-p. (a) Give p > 0 and e> 0, show that x-1-e 〈 fe(x) < x-1 forsufficiently large (b) For which p does J2cb(x)dx converge.
6. For p > 0, let fe(r) = r-ı (logr)-p. (a) Give p > 0 and e> 0, show that x-1-e 〈 fe(x)
2. Show that P[AIB] satisfies the three axioms of probability b) PISIB] 1 for sample space S c) If AnC 0 (empty set), then P[An CIB] P[AIB] + P[CIB]
2. Show that P[AIB] satisfies the three axioms of probability b) PISIB] 1 for sample space S c) If AnC 0 (empty set), then P[An CIB] P[AIB] + P[CIB]
a. Define what it means for two logical statements to be equivalent b. If P and Q are two statements, show that the statement ( P) л (PvQ) is equivalent to the statement Q^ P c. Write the converse and the contrapositive of the statement "If you earn an A in Math 52, then you understand modular arithmetic and you understand equivalence relations." Which of these d. Write the negation of the following statement in a way that changes the...
Exercise 1. Let f : R R be differentiable on la, b, where a, b R and a < b, and let f be continuous on [a, b]. Show that for every e> 0 there exists a 6 > 0 such that the inequality f(x)- f(c) T-C holds for all c, x E [a, 히 satisfying 0 < |c-x| < δ
If P(E)9 and P(F)-.8, show that P(EnF)2.7. I inequality, namely, n general, prove Bonferroni s Use induction to generalized Bonferroni's inequality to n events and show the result.
For the given function f(x) and numbers L, c, and e > 0, find an open interval about c on which the inequality fx)-L<e holds. Then give a value for 6 > 0 such that for all x satisfying 0< x -cl <8 the inequality f(x)-LIe holds. f(x) mx, m> 0, L = 2m, c 2, e = 0.03 For what open interval does the inequality f(x)-L hold? <e (Type your answer in interval notation.) Find the largest value 6>0...
Question 3 (Chapter 6) 12+3-3-6 14 marks] Fix p E N and consider the following set: (c) Compute Ci and C2 (d) Show that a 0 is an extreme point of Cp. Hint: you may use (without proof) that the family of functions fe',e", ..., e) is linearly independent.
Question 3 (Chapter 6) 12+3-3-6 14 marks] Fix p E N and consider the following set: (c) Compute Ci and C2 (d) Show that a 0 is an extreme point of...
3. (25 pts) Let fe C2[a, b], for a < b, and let {p,}0 be Newton's method, where p,n E [a, b] for all n 2 0. Suppose pn Converges top E [a, b], where f(p) 0, f'(p) 0, and p #p for all n 2 0. Find an expression for X 2 0, where sequence generated by a. Pn+1 -p lim = Pn-pl2
3. (25 pts) Let fe C2[a, b], for a
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