2. Consider the two signals X1(t), and X2(t) shown below, sketch (a) S(t) = X1(t) + X2(t) X1 (t) (b) M(t) = X1(t).X2(t) (c) D(t) = X2(t) – X1(t) x2 (t) -10 10 3. 3.

Q: Q3: Prove that Σ(y₁ - y) = 0 AND Σfi(y₁ - y) = 0

A: The question is about proving given equation. To prove : ∑ ( yi - y¯ ) = 0∑ fi ( yi - y¯ ) = 0

Q: 1. Show that S, has the presentation (x, y|x² = y = (xy)² = 1). %3D

A: The symmetric group S3 is given as S3=I, 1 2, 1 3, 2 3, 1 2 3 1 3 2 where  I=123123, 1 2=123213, 1…

Q: a of the following juncions. s- 1 (b))F(s)% = %3D 2s2 -8 s + 11

A:

Q: Given u(x, t) = t-²e¯, 2 , compute:

A: We can find the required answer as below.

Q: a²z If z = In(x² + y²)., find ax2 ay?

A: Here we have;  z=ln(x2+y2) ...................(1) partial derivative with respect to x

Q: graph the points z = x + iy that satisfy the givenconditions.a. | z | = 2 b. | z | < 2 c. | z |…

A: a. We have given z=2 Then, put z=x+iy This implies that x+iy=2x2+y2=2x2+y2=4 Therefore, x2+y2=4…

Q: Let a e R" and let A = aaT. Show that xT Ax = (a" x)².

A:

Q: verify that (A + B)T = AT + BT

A:

Q: Show that the following relationships hold. a) (1+i)™sn[ = Sn+m[ ¯ Sm| b) än+1] = 1+ an] (m) (m), c)…

A: Considering the LHS (Left hand Side) : LHS=(1+i)mSn|¯ RHS=Sn+m|¯-Sm|¯ ( Right Hand Side)…

Q: 8 15 Va 8 dx Evaluate 3 5 12 x 4 – In(x®)+12 x 3 + C, Where C is the parameter 5 - - 12 x 4 –…

A:

Q: show that Ф (x, y, z) = аф PDE : агф дуг + дуг I х2 +у2 +Z2 024 072 satisfies the

A:

Q: 3. Show that the action of C4(2) operator gives Ca(2) (X1, Y1, Z1) = (-yı,X1, Z1)

A: Consider this four-fold rotation around the z-axis. The effect of this operation on x,y,zis given by…

Q: 4. Evaluate both sides of Green's theorem, for: F = (-x²y,xy²) where C is shown in the following…

A:

Q: Q3: a) Let ƒ : Z → Z be such that f(x) = Vx² – x + 1. If invertible, and if it is, what is its

A:

Q: 10.Show that Φ(t) = Ψ(t)Ψ−1(t0), where Φ(t) and Ψ(t) are as defined in this section.

A:

Q: 2- Sketch the signal x(t) = u(t +1) + 2u(t) –u(t –1) –u(t - 2) -u(t-3)

A: Given signal is x(t)=u(t+1)+2u(t)-u(t-1)-u(t-2)-u(t-3) For signal u(t+1), graph is For signal…

Q: 4 16 Define T: R? - R° by T(x) = Ax, where A = 1 3 -

A:

Q: Q4 Prove that E(",") = ("::)- () .n 2mzr. %3D +1 +1

A: we have to prove that                                                   ∑k=0mn-kr=n+1r+1-n-mr+1 now,…

Q: 2. A firm produces two goods in pure ompetition & has the pllowing TR & TC functions R= 15Q…

A: In this question, concept of optimization is applied. Optimization Mathematical optimization is the…

Q: 3. If Rw (T) is the autocorrelation of a white noise w(t) and S e-217| Rw(t)dt = 1, find Sw(f).

A:

Q: Find (T3 • T, o T1)-1 if Ti 1(х, у, 2) %3D (х - у, — у, х+2) T."(х, у, 2) %3D (х -у+z, 0, 3x) T(х,…

A: This question is related to linear transformation . See 2nd step.

Q: Четверг show that the inu of a bounded E tix-oy ne 7 8. use T

A:

Q: Show that E(SSE) = (n - k-1), as in Theorem 7.3f, using the following approach. Show that SSE = y'y…

A: Please find the explanation below. Thank you

Q: 28. (a) Prove that for x e R" and s < t, Ds(x) C D;(x).

A: The notation Dsx represent  an open ball of radius s, centered at x∈ℝn. Let x=x1, x2, …, xn,…

Q: If z =e* + In(xy) Evaluate- ,and ,at the point (1, 2). Hence, calculate the change in z when x…

A:

Q: Which of these best describes T?

A:

Q: If f(t)dt = 1, S g(t)dt = 3, evaluate (3f(t) + 2g(t) – 5)dt.

A:

Q: 2): Consider the DE y" – y" – 6y = 0 -

A: Follow the procedure given below.

Q: Verify that u' (t – a) = 8(t – a) by solving the problem x' = 8(t – a); x (0) = 0 to obtain x (1) =…

A: What is Laplace Transformation: Laplace transformation is crucial for control system design. Laplace…

Q: LZHP - DTNV. Complete the statement below. ZZ = _2_

A: Consider the following condition: LZHP~DTNV Hence, ∠L≅∠D∠Z≅∠T∠H≅∠N∠P≅∠V

Q: two variables are independent, each with the pmf given in the accompanying table (so X,

A:

Q: H.W. a PDE from the followi'ng- @u=f() Form メツ 0u=fcx?ー)

A:

Q: The linearization at a = 0 to √8+7e is A + Ba. Compute A and B. A B= || ||

A:

Q: T: M22 ---&gt;M22 defined by T(A) = AB,

A: Given, a matrix B, and defined a map T(A) = AB, where T: M2×2  ---&gt; M2×2. We have to find the…

Q: Find the Ker(T) and Rng(T) and their dimensions. T:R →R defined by T(x)= Ax, where 1 -1 27 -3 3 -6

A: The objective is to find ker(T) and range (T) and their dimension.

Q: Find the F.T. of the following-peridic signal -2 2.

A: From the given graph  periodic function is fx=1-x, 0≤x≤1 with period 1,

Q: graph the points z = x + iy that satisfy the givenconditions. | z |&gt; 2

A: Complex numbers

Q: 4. Prove that (À × B) · (B x T) × (T x A) = [ABT)

A: The objective is to prove that

Q: #6

A: A function T; Rn → Rm is called a linear transformation if T satisfies the following two linearity…

Q: 4. Using the chain rule, find the following A. , if z = u³v²w² and u = t3, v = 3t – 6, and w = t2…

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

Q: 4. Now, determine the value of c in (0,1) such that f'(c) = 2 fx) =4x

A:

Q: In C[0, 1], are the functions {x2 , x3 , e2x} linearly independent?

A: Linearly dependent and independent functions: Two differential functions f(x) and g(x) are said to…

Q: 2. Use a theorem to find the ff. Zx if yz? + z = 5xy – 1.

A: Find out the zx when function is given as yz2+z=5xy-1 or yz2+z-5xy+1=0

Q: 2/ Compute fc E.dr and Where F Gxry2=(3+ 12x²,3xy²2 is a line segment from Co,01 to (4,2) %D Fdr =

A:

Q: Show that (A − B)T = AT − BT .

A: We have to use the properties of subtraction of matrices and matrix transpose operation. In matrix…

Q: If θ̂ is a point estimator of θ, prove that MSE(θ̂) = V(θ̂) + (B(θ̂))2

A: We have given that θ̂ is a point estimator of θ. We know that  MSE(θ̂) = E((θ̂ - θ)2 )

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!