Q.Let X = {m, n, p, q}, and E= {0, X, {m}, {n}, {p, q}, {m, n}, {n,p, q}, {m, p, q}}. Then > is a sigma algebra over X. а. b. Σ is not a sigma algebra over X >- {m, p, q} a sigma algebra over X d. None of these. a b C O d

Q: Q. If E, and E, are segma algebras over X, Σ EUL, is a sigma algebra over X а. '2 b. n2, is a sigma…

A:

Q: An example of a partition of the integers is E Z}

A: Set of integers is defined as {... -3, -2, -1, 0, 1, 2, 3,....}. A partition of a set is defined as…

Q: Let A = {4n^2 + 1|n ∈ Z}. Show that A is a subset of Z but not equal.

A:

Q: Q. If E, and E, DU is a sigma algebra over X Σ are segma algebras over X, a. (2 b. En E, Σ is a…

A: b. ∑1∩ ∑2 is a sigma algebra over X

Q: Q. If E, and E, are segma algebras over X, 2, is a sigma algebra over X a. (2 b. ,n, is a sigma…

A:

Q: 3. If x is uncountable, then prove that A = {ECX: ECX: algebra O X: E is countable or E² is…

A:

Q: Recall that we defined the Borel o-algebra in R", to be the o-algebra generated by the collection of…

A: Since you have posted a question with multiple subparts, we will solve the first  two subparts…

Q: 3. Let = {1,2,3,4,) and C= {{1,2,3}, {4}}. Derive the sigma-algebra generated by C, i.e. that…

A: Given: Ω=1,2,3,4 C=1,2,3,{4}Find:Derive the sigma-algebra generated by C.

Q: 1. For every element x in A, there is an element y in B such that (x, y) E F.

A:

Q: Q2. (a) Prove that if , and E, are segma algebras over X, then . Σ is a sigma algebra over X. (b) Is…

A:

Q: Let G = U(15) and N = U,(15).If U(n) = {x E U(n) such that x mod k = 1}, then the factor group G/N…

A: We have to solve given problem:

Q: Suppose that {$1,...Øn} F p and I F ø; for all i < n. Show that I F þ.

A: Proof: Suppose ϕ∈Γ∗ϕ∈Γ∗. As Γ∗Γ∗ is maximally consistent it clearly follows that there exists a…

Q: Let T E Sn be the cycle (1, 2, ..., k) e Sn where k < n. (a) For o E Sn, prove that oTo- (o(1),…

A:

Q: Q. If E, and , are segma algebras over X, 2uL, is a sigma algebra over X b. ,nE, is a sigma algebra…

A:

Q: Prove that if Σ1 and Σ2 are sigma algebras over X, then Σ1 ∩ Σ2 is a sigma algebra over X.

A:

Q: 1. Assume (X, o) and (Y, on X x Y as are groups. Let X × Y = {(x, y) |x € X,y E Y} and define the…

A:

Q: I denotes the set of real numbers and (*) is an operation on R such that a * B = a +B+ aß for ali a,…

A:

Q: 13. If T is self-adjoint, then S*TS is self adjoint for all S; if S is invertible and S*TS is…

A:

Q: For allne z, gcdin,nt)=l.

A:

Q: Prove that all elements in S are multiples of 3. by Structualindrction 3ES X,Y ES > (X y )ES

A:

Q: Let x E Q'. Show that for any n EN, there exists Xn E Q such that |x- Xal <

A: Given x∈ℚ', that x is a fixed irrational number. We know that between any two real  number, there…

Q: normal sibgroup of Exer cise 3, Let N Re G. Prove that [N,NJis a norumal a a silgrep 4 G group

A: A subgroup H of group G is called a normal subgroup of G if ghg-1 ∈ N , for every g ∈ G , h ∈ H. The…

Q: Let 2=(1,2,3,4,} and C={{1,2,3}, {4}}. Derive the sigma-algebra generated by C, i.e. that smallest…

A: Given set is Ω={1,2,3,4} where C={{1,2,3},{4}} The objective is to derive the sigma algebra…

Q: 3. Find the spectrum of the following operator A: ( (: A(x1,..., Tn,..) = (x1 + x2, x2 + x3, ..., Xn…

A: As per the question we are given a linear operator A from l2 to l2  And we have to find the spectrum…

Q: 0/ The center of agroup G is denoted by xcG yone of the following Statements is false: (Sn)={e} for…

A: Let G be a group and its center is denoted by ZG. It is given by ZG=x∈G:ax=xa, ∀a∈G that is…

Q: ) Given the set S = {1,2,...,n}, show that Sym(S) is Abelian for n = 1, 2, and non- Abelian for n>2,…

A: This is a problem with the theory of Groups.

Q: In a group (G, 0) the elements a and b commute. Also O (a) = m and O (b) = n, are prime to each %3D…

A:

Q: Let H x K = { (h,k) | h in H, k in K } such that (h1,k1) + (h2,k2) = (h1 + h2, k1 + k2), for all h1,…

A:

Q: Q. If >, and E, are segma algebras over X, 12 Eu, is a sigma algebra over X а. '2 b. ,n, is a sigma…

A: Correct answer is option B Explanation is given below

Q: Exercise 14.5.3. Using “switchyard", we proved that S, is generated by the permutations (12) and…

A:

Q: QLet X = {m,n, p, q}, and = {6, X, {m}, {n}, {p,q}, {n, p, q}, {m, p, g}}. Then is a sigma algebra…

A: Let X=m,n,p,q and ∑=ϕ, X, m, n, p,q, n,p,q, m,p,q

Q: Let n be a positive integer, and let nZ = {nm|m ∈ Z}. (a) Show that ⟨nZ, +⟩ is a group. (b) Show…

A: It is given thatnZ=n m |m ∈ ZWe have to show (nZ,+) is a groupnZ=n m |m ∈ ZClosure property :- Let…

Q: Q. If E, and E, a. E,n E, is a sigma algebra over X b. EuE, is a sigma algebra over X E, E, is a…

A:

Q: Q. If 2, and >, are segma algebras over X, is a sigma algebra over X (2 а. (2 b. ,n, is a sigma…

A:

Q: > a sigma algebra over X ?

A: Hello. Since your question has multiple parts, we will solve first question for you. If you want…

Q: Which of the following statements involving sigma notation is always true? Ο Σακ- b) = Σ b-Σα ) k=1…

A: We can check which statement is correct.

Q: 1. Let 3(0,7) u {7 + Define < on Sby a <b if b-a ES. Show < is not an order on n=1 S.

A:

Q: 1\Find the inverse for each element in the following mathematical syste A) (Z, *) where defined as…

A: according to our guidelines we can answer only three subparts, or first question and rest can be…

Q: Q2. (a) Prove that if), and >, are segma algebras over X, n, is a sigma algebra over X. then a sigma…

A:

Q: 2. Every group of index 2 is normal.

A: Given :   Every group of index 2 is normal

Q: IV.Let Ω {a,b,c,d,e,f.g) and = {{a.f.g}, {d,e,f}, {b,f}} Construct the minimal sigma-field…

A:

Q: Q. If E, and , are segma algebras over X, EuE, is a sigma algebra over X b. E E.E, is a sigma…

A: Given that, correct answer is option B Explanation is given below

Q: Q. If 2, and , are segma algebras over X, En E, is a sigma algebra over X b. E,uE, is a sigma…

A: Given problem is :

Q: Let N = [0, 1]. Determine the smallest sigma algebra on N containing 0, ) and 0, 4

A: Given: Ω=0,1 and the sets [0,1/4) and [0,1/2) To find: Smallest sigma algebra on Ω which contains…

Q: and are groups. Let (2, 9)| E A, y E. and denne the operation on X x Y as (T1, 41) * (x2, Y2) = (x1…

A:

Q: Prove that : a"+ (a+ 1)° =1 +a(1+い) (mod a la+s) for all a, m,n E Z.

A:

Q: Σ Q. If E, and E, E,n£, is a sigma algebra over X b. E,uE, is a sigma algebra over X .E,\E, is a…

A: The solution is given by using the definition of sigma algebra as follows

Q: I need help with # 18 attached for abstract algebra plz. can u list steps clearly so I know how do…

A:

Q: Prove that for any k > 0, nk e o(2v™).

A: The complete solutions are given below

Continuous Probability Distributions

Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.

Normal Distribution

Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!