Homework Answers
By definition A collection of sets forms a partition of the sample space if the sets (i) are mutually exclusive and (ii) exhaust the entire sample space.
Then only A and C are the properties of a partition. The correct option is F.
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1.Let Ai, A2,. . , Ak be collection of sets that partitions the sample space S....
Part(c) and partd(d) 1.34 Let A. A.. be a countable collection of subsets of U.Prove the...
Part(c) and partd(d)
1.34 Let A. A.. be a countable collection of subsets of U.Prove the following. a) If B cU, An, then B U,(An n B). b) If Un An-U, then E U(An E) for each subset E of U c) If Ai, A2,... are pairwise disjoint, so are Ain E, A2n E,... for each subset E of U d) We say that Ai, A2,... form a partition of U if they are pairwise disjoint and their union is...
1. (10 marks) (a) Let m events Bi, , Bm form a partition of the sample space Ω and let event A be...
1. (10 marks) (a) Let m events Bi, , Bm form a partition of the sample space Ω and let event A be any event such that A c S2. Then show that given Bi > 0 for j = 1,.. ., m (b) Considcr a clinical trial where group of paticnts arc trcated for depression. As in many such trials a patient has two possible out- comes, in this study a relapse and no relapse. Refer to a relapse...
1. Let sample space S={1, 2, 3, 4, 5, 6}. Is it this a true or...
1. Let sample space S={1, 2, 3, 4, 5, 6}. Is it this a true or false statement : event C{1,2,3} and D={4,5,6} are both mutually exclusive and collectively exhaustive? My teacher said this is True please explain why in a picture and explanation 2.Let sample space S={1, 2, 3, 4, 5, 6}. Is it this a true or false statement : event E={3,4,5} and F={1,2,3} do not form a partition? My teacher said this is True please explain why...
A2. (a) We roll a fair die twice. Describe a sample space to model this experiment....
A2. (a) We roll a fair die twice. Describe a sample space to
model this experiment.
C4. Consider Problem A2 (a) in Homework 1. Suppose that all the outcomes in the sample space are equally. Let Ai be the event that the sum of the two numbers is greater than 9. Let A2 be the event that both numbers are identical (a) Construct a probability model for this experiment (Specify the general (b) List the outcomes in event A1, and...
#5 (4 pts.) Consider the following sample space S and events A and B. s-(-4 <...
#5 (4 pts.) Consider the following sample space S and events A and B. s-(-4 < x < 2, 6 < x < 12), A={-4 < x < 0}, B=(-1 x<2), A and B are: a. (mutually exclusive, independent) b. (mutually exclusive, dependent) c. (non-mutually exclusive, independent) d. (non-mutually exclusive, dependent) #6 (4 pts.) In problem #5 P(B-A)- c. 1/4 d. 1/6
2. Consider tossing a coin twice. Denote H ="head" and T ="tail" (a) List all outcomes...
2. Consider tossing a coin twice. Denote H ="head" and T ="tail" (a) List all outcomes in the sample space S (b) Let X count the number of heads. List all outcomes in the events Ao = {X = 0}, Ai = {X=1 and A2 {X = 2}. Are all the events Ao,A1,A2 mutually exclusive? Explain. (c) Suppose P(H) = 0.6. Find the probability mass function of X: f(x) = P{X =x} (d) Find the cumulative distribution function of X:...
1. Let (S;F;P) be a probability space with A 2 F and B 2 F such...
1. Let (S;F;P) be a probability space with A 2 F and B 2 F such that P(A) = 0:3 and P(B) = 0:4. Find the following probabilities under the specified conditions. Note that I don’t expect you to have to show much work in answering this question. (a) either A or B occurs if A and B are mutually exclusive (b) either A or B occurs if A and B are statistically independent (c) either A or B occurs...
are even, r even. A 1.1-14·Let the interval [-r,r] be the base of a semicircle. ne...
are even, r even. A 1.1-14·Let the interval [-r,r] be the base of a semicircle. ne of the f a pot is selected at random from this interval, assign abilty of a probability to the event that the length of the perpen- e slot into dicular segment from the point to the semicircle is less than r/2. 1.1-15. Let S = A1 U A2 U U Am, where events (a) If P(A1) P(A2)P(A), show that P(Ai - (b) If A...
(1) Let a (.. ,a-2, a-1,ao, a1, a2,...) be a sequence of real numbers so that f(n) an. (We may eq...
(1) Let a (.. ,a-2, a-1,ao, a1, a2,...) be a sequence of real numbers so that f(n) an. (We may equivalently write a = (abez) Consider the homogeneous linear recurrence p(A)/(n) = (A2-A-1)/(n) = 0. (a) Show ak-2-ak-ak-1 for all k z. (b) When we let ao 0 and a 1 we arrive at our usual Fibonacci numbers, f However, given the result from (a) we many consider f-k where k0. Using the Principle of Strong Mathematical Induction slow j-,-(-1...
Question 3: Eigenvalue Theory 1 (a) Let A e Cnxn, and let (Ai, an), (Ak,Xk) be...
Question 3: Eigenvalue Theory 1 (a) Let A e Cnxn, and let (Ai, an), (Ak,Xk) be eigenpairs where all λί are distinct. Show that the corresponding eigenvectors r1,. .. Tk are linearly independent. (b) Let A, B e C"xn be similar. Show that A and B have the same char- acteristic polynomial, same eigenvalues including algebraic and geometric (c) Do A and B fro (b) share the same singular values? Justify.
Part(c) and partd(d)
1.34 Let A. A.. be a countable collection of subsets of U.Prove the following. a) If B cU, An, then B U,(An n B). b) If Un An-U, then E U(An E) for each subset E of U c) If Ai, A2,... are pairwise disjoint, so are Ain E, A2n E,... for each subset E of U d) We say that Ai, A2,... form a partition of U if they are pairwise disjoint and their union is...
1. (10 marks) (a) Let m events Bi, , Bm form a partition of the sample space Ω and let event A be any event such that A c S2. Then show that given Bi > 0 for j = 1,.. ., m (b) Considcr a clinical trial where group of paticnts arc trcated for depression. As in many such trials a patient has two possible out- comes, in this study a relapse and no relapse. Refer to a relapse...
A2. (a) We roll a fair die twice. Describe a sample space to
model this experiment.
C4. Consider Problem A2 (a) in Homework 1. Suppose that all the outcomes in the sample space are equally. Let Ai be the event that the sum of the two numbers is greater than 9. Let A2 be the event that both numbers are identical (a) Construct a probability model for this experiment (Specify the general (b) List the outcomes in event A1, and...
#5 (4 pts.) Consider the following sample space S and events A and B. s-(-4 < x < 2, 6 < x < 12), A={-4 < x < 0}, B=(-1 x<2), A and B are: a. (mutually exclusive, independent) b. (mutually exclusive, dependent) c. (non-mutually exclusive, independent) d. (non-mutually exclusive, dependent) #6 (4 pts.) In problem #5 P(B-A)- c. 1/4 d. 1/6
2. Consider tossing a coin twice. Denote H ="head" and T ="tail" (a) List all outcomes in the sample space S (b) Let X count the number of heads. List all outcomes in the events Ao = {X = 0}, Ai = {X=1 and A2 {X = 2}. Are all the events Ao,A1,A2 mutually exclusive? Explain. (c) Suppose P(H) = 0.6. Find the probability mass function of X: f(x) = P{X =x} (d) Find the cumulative distribution function of X:...
are even, r even. A 1.1-14·Let the interval [-r,r] be the base of a semicircle. ne of the f a pot is selected at random from this interval, assign abilty of a probability to the event that the length of the perpen- e slot into dicular segment from the point to the semicircle is less than r/2. 1.1-15. Let S = A1 U A2 U U Am, where events (a) If P(A1) P(A2)P(A), show that P(Ai - (b) If A...
(1) Let a (.. ,a-2, a-1,ao, a1, a2,...) be a sequence of real numbers so that f(n) an. (We may equivalently write a = (abez) Consider the homogeneous linear recurrence p(A)/(n) = (A2-A-1)/(n) = 0. (a) Show ak-2-ak-ak-1 for all k z. (b) When we let ao 0 and a 1 we arrive at our usual Fibonacci numbers, f However, given the result from (a) we many consider f-k where k0. Using the Principle of Strong Mathematical Induction slow j-,-(-1...
Question 3: Eigenvalue Theory 1 (a) Let A e Cnxn, and let (Ai, an), (Ak,Xk) be eigenpairs where all λί are distinct. Show that the corresponding eigenvectors r1,. .. Tk are linearly independent. (b) Let A, B e C"xn be similar. Show that A and B have the same char- acteristic polynomial, same eigenvalues including algebraic and geometric (c) Do A and B fro (b) share the same singular values? Justify.
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