4 Let (2, A, P) be a probability space, let X be an integrable randomvariable. Let C be a sub-o-algebra ofA. Prove that|(E(X|C)| < E( |X| | C ), P -a.s.

Answered: Let (2, A, P) be a probability space,…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter8: Polynomials

Section8.3: Factorization In F [x]

Problem 22E: Let ab in a field F. Show that x+a and x+b are relatively prime in F[x].

See similar textbooks

Similar questions

Let ab in a field F. Show that x+a and x+b are relatively prime in F[x].
Consider the probability space (Ω,A , P) where Ω = R and A is the Borel σ-algebra on R. Suppose that for any n = 1, 2, . . . , we have P((−∞, 2−n]) = 1/2 and P((−∞, −2−n]) = 1/3 − 1/(4^n) . Given the above, compute the following probability values, if possible, showing complete justifi- cation for every step. If it is impossible to compute a value precisely, provide the tightest possible bounds on it. P((−∞, 0]) P({0}) P((−∞, −1]) P((1/4, π/7]) limn→∞ P((0, n])
Let A be a non-empty and bounded subset of R, and let x_0=supA. Prove that x_0 ∈ A or that x_0 is an accumulation pt of A.
Prove the following generalization of Theorem 3: IfX1, X2, ..., and Xn are independent random variables andY = a1X1 + a2X2 +···+ anXn, thenMY(t) = ni=1MXi(ait) where MXi (t) is the value of the moment-generating func-tion of Xi at t.
Let X = (X1, X2, ..., Xn)T be a random vector with mean vector μ and covariance matrix Σ. Suppose that for every a = (a1, a2, ..., an)T ∈ Rn, the random variable aTX has a (one-dimensional) Gaussian distribution on R. a) For fixed a ∈ Rn, compute the moment generating function MaTX(u) of the random variable aTX, writing the answer using μ & Σ. b) Define MX(t1, t2, ... , tn), the moment generating function of the random vector X, and express it in terms of MtTX, the moment generating function of tTX where t = (t1, t2, ..., tn). c) Combining a) and b), compute the moment generating function MX(t) of X and hence prove that the random vector X has a Gaussian distribution.
3. Let Ω = [0, 1] and A = B(Ω) (i.e., the Borel σ-algebra generated by [0, 1]). Suppose (Ω, A, P) is a probability space with a probability measure P, where P([a, b)) = b − a, ∀a, b ∈ Ω. For example, P ( [0, 1/2)) = 1/2. Find P({0}) and prove it.
Let A be a denumerable set. Prove that A has a denumerable subset B such that A − B is denumerable.
4.3.2 Let Y ∼ Uniform[0, 1], and let Xn = Yn. Prove that Xn → 0 with probability 1.
Let X = (X₁, . . . , Xn) denote an (n × 1) vector of independent random variables Xi∼ N (0, 1), i = 1, . . . , n. Let A = [aij] denote an (m × n) matrix, and define a random vector Y = (Y₁, . .. , Ym) asY = AX. Show that Cov(Yi, Yj) Which of the following lines is a valid argument? See attached pic
Let a ≠ b in a field F. Show that x + a and x + b are relatively prime in F[x].
Let a ∈ G be such that |a| = ∞ . For m≠n prove that a^m ≠a^n
Let D be a subset of R. Let x0 ∈ D such that x0 is not an accumulation point of D. (a) Use the formal negation of the definition of “accumulation point” to prove that there exists a µ > 0 such that (x0 − µ, x0 + µ) ∩ D = {x0}. (b) Let f : D → R. Prove1 that f is continuous at x0.

Recommended textbooks for you

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

Linear Algebra: A Modern Introduction

Algebra

ISBN:

9781285463247

Author:

David Poole

Publisher:

Cengage Learning

Algebra & Trigonometry with Analytic Geometry

Algebra

ISBN:

9781133382119

Author:

Swokowski

Publisher:

Cengage

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

Linear Algebra: A Modern Introduction

Algebra

ISBN:

9781285463247

Author:

David Poole

Publisher:

Cengage Learning

Algebra & Trigonometry with Analytic Geometry

Algebra

ISBN:

9781133382119

Author:

Swokowski

Publisher:

Cengage

SEE MORE TEXTBOOKS

Let (2, A, P) be a probability space, let X be an integrable random variable. Let C be a sub-o-algebra of A. Prove that |(E(X|C)| < E( |X| | C ), P -a.s.