(a) Explain why X meets the criteria to be a discrete r.v., and give its space. (b) Find the pmf of X.

Q: If llall is a vector norm for a convex XEIR show that f(x)=11x11 is function.

A:

Q: How do you know that for all a, b, c, d: ax + bx will be orthogonal to cx? + d ?

A:

Q: The set Qn (0,2) has Lebesgue measure 0.

A:

Q: Find an example of a bounded convex set S in R2 such that its profile P is nonempty but conv P ≠ S.

A:

Q: Given points pı = , and p3 in R², P2 let S = conv{pj,P2»P3}. For each linear functional f, find the…

A:

Q: Prove that the set A = {x ∈ R | 3x + 2 ≤ 0} has a maximum

A:

Q: Let XC Rn convex. Prove that x is an extreme point of X if and only if the set Cis convex, where X =…

A: Let X⊂ℝn be convex . To prove: x is an extreme point of X if and only if the set C is convex, where…

Q: If X/M ={ [x], xe Xx} show that X/M is a linear space, Where [x] is equivalence class

A: To show that X/M is a linear space, we need to verify that it satisfies the following axioms:…

Q: Find the locus of the points z that satisfies 3(22 + z?) + (8 – 16i)z + (8+ 16i)z – 10|z|? = 16.

A:

Q: the quotient space of a T0−space necessarily T0

A:

Q: (a) Determine whether or not it is convex or concave. A = {(x, y) = R²|yx4, and -2≤ x ≤ 2} B = {(x,…

A: Necessary Sufficient Nature f'x0=0 f'x0=f''x0=f'''x0=..=fn-1x0=0  fnx0>0 n even Convex f'x0=0…

Q: Show the 3-dimensional ball B₁: ((..) ER³ | x² + y² +2² 0.

A:

Q: (d) Show that T is bounded and find ||T||. (e) Find the adjoint T* of T.

A: We will use the vector/matrix notations to visualise things easily.

Q: Find the dimensions of the following linear spaces. (a) RSX3 (b) P6 (c) The real linear space C

A: In general Rm×n is a vector space of dimension m×n. So, Dimension of R5×3 is 5×3 = 15 (a) 15

Q: By using the fact that P[AU B] 1 –Ei=, P[A£].

A: (a). Prove that P [A U B U C] ≤ P[A] + P[B] + P[C]: It is given that P [A U B] ≤ P[A] + P[B]. The…

Q: Find the null space for A. = [1 null(A) A = 15-5 0 1 -³] 3 = span

A:

Q: Find the norm of f(x) = √√1-x² on the interval [- 1,1]. 2√3 3 O O O W|N H

A:

Q: If A is bounded below and B is bounded above, show that inf(A − B) = inf A − sup B

A:

Q: Let g : X c R"→ R be a function defined on the finite set X. Does g achieve a maximum and/or a…

A:

Q: Let B be the set of all bounded sequences of real numbers and define the function d: B xB +R by d(x,…

A: In the above question it is given that B is the set of all bounded sequences of real numbers and a…

Q: a) Find a Lagrange interpolating polynomial in the form of P, (x) = ax³ + bx² + cx+d befitting the…

A:

Q: Let A = {AC R| A is countable} and A2 = {A C RI A° is countable} which is a measurable space? (IR,…

A:

Q: Find the supremum and infimum of the set {x € R|10/T – I > 0}, if they exist. If they exist, provide…

A:

Q: Extend the linearly independent Set S={x+1, 2X²} to the space base P₂ (R)

A: We are given a linearly independent set S= {x+1, 2x2} We have to extend S to basis of P2(R).    Let…

Q: (b) Let X = {0, 1} and let (X*,d) be metric space, where d(x, y) = the number of elements of the set…

A:

Q: Theorem 4 If B CA, P(B)< P(A).

A:

Q: (1 x x²). Let F be the 2: Consider the polynomial space P₂ with the standard base V = corresponding…

A:

Q: What is the dimension of the following linear space? Pn Note: Pn is the set consisting of the zero…

A:

Q: Describe the set {TER²: doo (r,(0, 0)) = 1}.

A: To Describe: the set {x∈ℝ2: d∞(x,(0,0))=1}.

Q: If f (x) = and 8(x) = 2vx – 5, determine f (g(x)) and g(f(x)). 2x2

A:

Q: Let v= C[-1,1] What is the norm of the functionf(x) = - x in this space? O A) 2V5 O B) 2V10/5 OC)…

A: An inner product in the vector space of continuous functions in [-1, 1], denoted asV=C-1, 1, is…

Q: a function f is continuous at x0 element of X if and only if f is both lower semi continuous and…

A:

Q: Suppose that X is a subset of Y, such that P(Y)>0. Is true or false that P(X|Y) is equal to…

A: It is given that X is a subset of Y This means that PX∩Y=PX...(i) The formula for PX|Y is given by:…