9. Prove or disprove. (a) Vn e Z*, n² < 2". (b) Vm e Z, if m is odd, then m² is odd.

Q: 2. Golve the pollowing uaing methods for non- exad ea vations, Jintar eouaton or Bernonlli's es…

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Q: 2. Using OLS or any other suitable method, Prove that E(ß1) = B1 E(ßo) = Bo

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Q: (a) Suppose X U (0,n), where n is a positive integer X-[X] U (0, 1). (b) Use the result of (a) to…

A: De Moivre's theorem gives an axiom for computing the powers of complex numbers. We first gain some…

Q: 12. Given: RX ITTW Prove RS ST = WS SX I

A: Find the solution below

Q: H) 9f zyzec then prove thal nl x=4=Z, (alogen) %3D

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Q: 4. Prove that for n > 0: E, 2* = 2"+1 – 1. sk3D0

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Q: 30) Let n 2 3 and k e Zn. Prove that in Dn. p u = up"-k, %3D

A: Note: There are more than one definition for dihedral group. The question is proved using a common…

Q: 5. prove thatS" erf (x) dx =o CA

A: ddx(erf(x)) =2 e-x2π   I =∫-aa erf(x) dxI= erf(x) x -∫ ddxerf(x) x dxI = x erf(x) -2π∫ x e-x2 dx-x2…

Q: 6. Prove that for x > -1 then Vn E N, (1 + x)" >1+ nx .

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Q: 6. Given: ZP=ZR; ZQ= ZS Prove: PO| RS P. 'R

A: ∠P≅∠R,∠Q≅∠S

Q: Prove for all mEw and n E w, we have that m+ ·n=m・n+n.

A: Given that m,n∈ω. To prove: m+· n=m·n+n

Q: 2xy + In(8 + z²), x 7v2 ', y = 9u + 3v, z = cos(4u). Find dwlðv when u = 5, v = 2. Suppose w = ,X =…

A: We have, w=2xy+ln(8+z2) x=7v2u,y=9u+3v,z=cos(4u). Point (u, v)=(5, 2)   As we can see that x, y and…

Q: Q1. Compute xdx x'+10 using trapezoidal and Simpson's 1/3 rules with the number of points 3 and 5.

A: The given integral is, I=∫01xdxx3+10 Dividing the interval (0,1)  into 2 intervals (for 3 points)…

Q: 1 prove that | e -ym dy = mr(m) -ут

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Q: 0/ the Valke of & that makes the Velteof the fumetion 2.

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Q: S Prove that L(cos 3t) (s²+9)

A: Given L(cos3t)

Q: 2) Prove that: d? L„0) = n;(n – 1) %3D dx²

A: Consider the given information: Let Laguerre polynomial is given by, Lnx=∑k=0n(-1)kk!nkxk…

Q: (2) O show that you Y2,43, and Yu are solutians to NOTE : Geueral Solufion

A: Thanks for the question :)And your upvote will be really appreciable ;) Solution is provided below,…

Q: 13 = 523411:KER 3is a sbuet 2x+1 wheher Bis 3 If B Decide t so is asbet of TR aboveand bounded find…

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Q: o prove that knVA=| Test the seuence (ConViordív) of (10) an = (^_-) ) an 15 ら 9n: 3ntl 3n-1 (の+ (7)…

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Q: Then, we assume that Wn+1 - H Wn - l kn+1 kin 1 2 E Wn-p Wn-p p=0 p=1 (6) n+1 - € Zn - € Vn+1 = 1 Vn…

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Q: Prove that Z(Sn) = {e} for n ≥ 3.

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Q: 2. Find +he source (メ) Q (x) that the So lutiom U (x it) of k + Qx) 2 - 0 l• Con e

A: given equation ∂u∂t=k∂2u∂x2+Qx,       -∞&lt;x&lt;∞           ...iux,0=0 claim- to find the source Qx…

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: eb Let A E M₁ (TR) be given If AAt = On HOME JATOT prove that A = On - Does this hold if A EMn (C)?

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Q: 1f itwas x25ect, yatunt prove Thut dy

A: Simple derivative

Q: Compute 32+n21-3n n=0 If this is divergent, input "O" as your answer If applicable, submit answers…

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Q: 9 Give with an not example of T: W≤X →Y, to XT Ext. .R. boundedag. - keše bes

A: Please find the solution innext step this is an non-example of Tietze extension theorem in Topology.

Q: S 1.Prove that L (coswt) = %3D s2+w2

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Q: If utt 4ura u(x, 0) = e", u4(x, 0) = 0, u, Ux → 0 as x → ∞, then u e* cosh(2t) O e2 cosh(4t) O e2¤…

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Q: O Find the set S on wlc fu x) = x Sin X convergas pointwise Cony

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Q: enoitetimil torte stigasb = 4", where m, neN. 10. Prove the identity: 3 r=0 itong s

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Q: Q3/ prove fry = fyx if f(x,y) = x + xy – 5x³ + In(x2 + 1)

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Q: 6. Which of the followg quti dy (a) de dy (b) de () +2ry- (e) (4) Al of the above

A: It is given that There are four differential equations option.

Q: 1. Prove with the usual notations that: (b) A + V =- (a) EV = VE = A (c) A= SEZ

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

Q: 1. Prove that L(t)=. 1 s2

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Q: - Pcosø prove that dw xy is Cont s Lony) - the fun ction (-1, 2) = x*y+z and x = PSinf cse y=pSi- -…

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Q: Determine Z- Trans furm in cludig vegiva of con vergence in n 7o メ(At n<o

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Q: (c) Evaluate d Gosh: where C: |z| = 2 by using Cauchy's residue theorem.

A: Given integral is ∮C dzcoshz;   C : z=2 As, we know  If z = x+ iy then z =x2+y2 As given that  z=2…

Q: Then, we assume that Wn+1 - H Wn - u kin+1 kn 1 2 E Wn-p E Wn-p p=0 p=1 (6) Zn+1 - € Zn - E Vn+1 Vn…

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Q: 2) Suppose S,T E L(V) are such that ST = TS. Prove that null S is invariant under T.

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Q: Find fCx) if flo)=0 and the tanGat (k,F CX) has ine af Slore x(x+2) a

A: Given for the function F(x), F(0)=0 and tangent line to F(x) at x,F(x) has slope xx2+23.

Q: 1 And the radios of Co for the pover Serie for the Power Serie vergence 00 xh (2"+1) (n?+1)

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Q: 12 Find the work done for a force F N from æ =2 to x =6 m. x2

A: Here we solve this by taking the integration with given limits

Q: Suppose w = In(æ² + y² + z²) , Compute dw/du and dw/dv in two different ways: ue3" sin(9v) , y :…

A: GIvenw=ln(x2+y2+z2) where x=ue3vsin(9v) , y=ue3vcos9v and z=ue3vTo find∂w∂u and ∂w∂v

Q: 5. If 4=2xy į +3x°y -3pyzk is solenoidal at(1,1.1). Find p.

A: DivA=0

Q: 2. Give the condition which ensure that |ez| < 1 where z in C.

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Q: 13. If w(x,y) = In(1+xy²), show that 2wxx + y³Wyx = 0. %3|

A: Given: w(x, y)=ln(1+xy2)  To show: 2wxx+y3wyx=0 Chain rule : ddx(f(g(x))=f'(g(x))g'(x) Quotient…

Q: 4: a) Let X.X., X -N (Hi,6f). Let Y,, Y2 Ym-N (u2,63). .... d (1-a) % CIfor oi when ai. a known.

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Q: 16.1. Use an antiderivative to show that for every extending from a point 21 to a point 22 2" dz =…

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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!