8. Let w, z EC such that |w| > |z|. Prove that (n+1) | ²7. < W for all n E N. |w| |w| - |z|

Q: Find P (-1.5 < or equal to Z < 0.2) where Z ~ N (0,1)

A: We will use Standard Normal table to find the required probability.

Q: show that a bounded Lineart on a complex Hilbert spase H is normal (i-eTT* = T*T) if and only if 11+…

A: We have to show that T  is normal if and only if ||T* x || = || Tx|| for all x € H. Using this, to…

Q: Let the vector v and the subspace W be as given below. 13 V = W = span -12 (a) Find a basis for W.…

A:

Q: Let z(x) ~N(0,1) Prop (x > -0.56) = Prop (x ≤ -2.17) = Prop (-1.56 ≤ x ≤ 0.72) =

A:

Q: -w #43 Let n», ², a; >0, 5=1,2,... and a₁ + a₂+...+ an = 1 Prove that a, a ₂ 2-92- + 4 + an 2-an 7…

A: We need to prove summation.

Q: prove for all x,y in R^n: (1+||x+y||) ≥(1+||x||)/(1+||y||)

A: Sol:- The inequality to be proven is: (1 + ||x + y||) ≥ (1 + ||x||)/(1 + ||y||) Let's start by…

Q: Let X~ N(2, 9). Compute P(1 ≤ X < 7).

A: From the provided information, Mean (µ) = 2 Variance (σ2) = 9 X~N (2, 9)

Q: Let f : N → P(N) be the function defined by f(n) = {n, n + 1, . . . , 2n} The following statement is…

A: 1) is false and 2) is true.

Q: Show that for all x, e* > 1 +x.

A: It is given that, ex≥1+xwe have to show that for all x.

Q: (a) Let the interval a, b be divided into n equal sub-intervals such that a = x0 < a1 < x2 < x3…

A: From this general formula we can obtain different integration formulae by putting n=1, 2,3 etc. We…

Q: Show that it is not true that for every error 1/m there exists n such that |Ek=1 polt – r/(1 - r)| <…

A: We will show that for every m there does not exist any unique n that works for all r in (0,1)

Q: XA and x & f(x)}. Show Cim ƒ.) 38. Let a, b, c, and d be any real numbers such that a < b and c < d.…

A:

Q: 2.59 Find || f |sup if a) f(x) = x on (-1, ). So if x E Q, b) f(x) = – x² .2 if x ¢ Q. -

A:

Q: Let x(k) = ku(k) h(k) = {0, ( 1, 0≤ k ≤ 4 elsewhere Let N = 11. (a) Plot x(k) for 0 ≤ k ≤ N-1. (b)…

A:

Q: 1. Let a be a positive real number and n be a natural number. Define S = {x €R: x >0 and x" a?…

A:

Q: 4.1 Prove that in any metric space (S, d) every closed ball S,[xo] is a closed set. 4.2 Let F be a…

A: Given (S,d) is a metric space. Three questions related to this are given. The solution to these…

Q: Suppose that Z~N(0,1). Then find Pr(2.78 < Z < 2.84)

A: Given data, Z~N(0,1) That is Mean=0 Standard deviation=1 P(2.78 &lt; Z &lt; 2.84)=?

Q: Suppose X is uniform on {6, 8, 10, 12, 14, 16}. What is Var[X]? A B C D E 27/12 33/12 35/12 35/3…

A: Solution-: Suppose X is uniform on {6,8,10,12,14,16} We want to find Var(X)=?

Q: 2- Find the set of lower bounds and the set of upper bounds of the following sets in Q, if it…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: 5. (8 points) Let S4be the set of all strings of 0's and 1's of length 4. Define F:S4 → Znonneg such…

A: The objective of the question is to understand the properties of the function F defined on the set…

Q: Let U= {x|x EN and x<7) A = {x|x is an even natural number and x<7} B=(x|x is an odd natural number…

A: Given problem:- Let   U={x|x ∈ N and x &lt; 7} A={x|x| is an even natural number and x &lt; 7}…

Q: 4x + 4 f(x) = {√ √²/√x + √// 0<x< 7/7/20 X 4 0 otherwise x < <0

A: To find the quantile and cumulative distribution function (CDF) for the given probability density…

Q: 1* Let X be the space of sequences of real numbers with only finitely many nonzero terms, with the…

A:

Q: :-(-1)" : n € N. State without proof the values of inf D (a) (i) Consider the set D = and sup D.…

A: As per the guidelines I am answering the only the part (a) of the question. For rest questions…

Q: If X is uniform on [-2,2] then p(x <3) = 3/4 O O o o

A: Given  X~Uniform(-2,2) Pdf of X is  f(x)= 1/4     , -2&lt;x&lt;2       = 0         , otherwise

Q: Prove that if a, b E Z - {0}, then gcd(a, b) = gcd(a, -b)

A: soln:gcda,b=Greatest common divisor of a and bEvery integer that divides a and bsimultaneously will…

Q: 15 Let (x+1) {x-2}" 20 g"k) = concave upX-) on what interval(s IS (NOte: Crinen gil but aske about…

A:

Q: (b) Show that xE R, x2 - 4x + 5 > 0.

A: As this is a multiple question type, only first question is to be solved, according to the Bartleby…

Q: (2.1) Let b> 0 and X [-b, b). Here we admit %3D B"open" in r if and only if (0} B or (-b, b) C B.…

A:

Q: If |Z| ≤ 1 and w) ≤ 1, then show that |zw|2s (1z|-|w|)² + (arg z - arg w)²2.

A:

Q: Show that p(y]x, z)p(x]z) p(y|z) p(x]y,z) = Show that P(x|z) = > P(x]y, z)P(y|z) %3D Show that…

A: If A and B two events then by definition of conditional distribution, p(A|B) = p(A∩B)p(B) We have to…

Q: 4. Show that In sin 6 d0 = – n 2 by applying the mean-value principle to In |1+ z| for |2| <r < 1,…

A: ∫0π2logsin(θ)dθ=12∫0πlogsinθdθ                         =12∫0π2logsin2θdθ…