Show that if P(A;) = 1 for all i > 1, then P(N1 A¡) = 1.

Q: Let S, := (1 - (-1)"/n :n e N). Find inf S, and sup S,.

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Q: Calculate M, for f(x) = 6 · In(æ³) over [1, 2]. M6 ||

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Q: Prove that if h E L'(R) and [ h= 0 for all s E R, then h = 0.

A: Use the Leibniz integral rule: For ∫axbx fx, tdt Where -∞<ax, bx<∞ We have, ddx∫axbx fx,…

Q: which of the folowiny -xト! is -3x 2 へ pove seies fore =e.e e E (-)"(-3x) n ハン 0 37 B. e (3x)" nこO *ス…

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Q: 0 <m < n such that am = e. (Hin

A: Subgroup generated by any element in a finite group will have at most n element where n is order of…

Q: Show that For all z E C, with z # 0, then: |z/²z=1 = z

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Q: evaluate nCx px qn-x  for n=5 x=2 p= 1/2

A: Given: n=5 , x=2 , p= 1/2 We need to find, nCx  px  qn- x We know that, nCx= n!/[(n-x)!(r)!] q= 1-p

Q: - Let X1, X2, ...,Xn~N(0,0) and T = Fi (T),Var(T)

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Q: What is the value of k such that P(z>k)=0.0630? Previous

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Q: Prove if equal to MPn (W) dx= 2" (n!)" (2n+)! that m is an integer

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Q: Show Sn+3 diverges by osing 6n +5 the test for divegence 1こV

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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: Suppose that x̅ = 12, Sx = 2, and z is 1. what is X?

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Q: s(2) f(z) is analytic in C

A: Ans : false

Q: Let c be a  positive integer, then (e^c+1)≥e^c for all c.T/F

A: Given that c is positive integer and we have to check whether the given inequality is true or not.…

Q: d x" = nx"-1 de 4. Prove that NɛQ

A: ddxxn=nxn-1 ;   n∈Q

Q: Q3: Prove that (9":n e Z} S {3":n Ee Z} but {9":n e Z} (3":n E Z}.

A: To prove:  {9n:n∈ℤ} ⊆{3n:n∈ℤ} but {9n:n∈ℤ} ≠{3n:n∈ℤ}

Q: Let a, b eR with a < b. Assume that a + <b - for all n e N. Show that n n=1

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Q: for n>2, 5 . Deermine for whpch n, n2-2 let |anlく,

A: Given  an = 5nn2-2

Q: 6. Use a substitution to find S (x* + 1)" z³dx.

A: Here we use substitution we find∫x4+111x3dxHere we substitute x4+1=du…

Q: Construct a formal proof of validity for (J A K) → L.

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Q: Let X and Y be independent and distributed as N(u, 1). Is T = X-Y sufficient?

A: According to factorization theorem, If the likelihood function L(x,θ) can be expressed in the form…

Q: Let P = {p:p=6x-1, for some x EZ} and Q= {q:6y+5, for some y EZ). Prove that P= Q

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Q: lar, let m = 2 and n e of A and B such that

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Q: ronskian of the DE : xp(x)y' + 3y = 0 = er² et? et?

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Q: Proof by contrapositive that for xy EZ, if xy is even then either x or y is even.

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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: ), P{|Zn – c| > ɛ} → 0 as n → o. Show that for any b on g, E[g(Z„)] → g(c) as n→o

A: *Answer:

Q: F(p), then prove that

A: simple use of integral transformation

Q: Let n be a positive integer, then n+1≥ for all n.T/F

A: We need to show that n+1≥n for all n where n is a positive integer. Lets proof by contradiction

Q: let x be have P.dif 0 < x <3 f(x)= ={th find The U.b. of P(IX-M73) and find The L. b.

A: Please see below

Q: Let Z ~ N(0, 1). Find P(Z > 1).

A: Z~N(0,1). This implies that Z is a Standard Normal Random Variable. The PDF of Z is given by,

Q: O N (AT) = R(A)-

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Q: Find the least value of “n" so that , k=n 1 199 Sn 2k-1 k=1 200

A: Given:          Sn=∑k=1n12k−1&gt;199200

Q: For any k = 1, 2, .... n, find E T₁+T2+ T₁+T2+ ... + Tk + In,

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Q: Gives a farme of v2 to within 10-3 Gives a farme of v5 to within 10-2

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Q: The characteristic of Zs[x]/ is 6. O None of the choices

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Q: Prove that x" dx „n+1 + C, n + 1 for n + -1.

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Q: 5. For non-negative integer n and r, where 1<rsn, show that (n-r+1)"C,1=n(""C,). r-1

A: Given n and r non negative integers with  1≤ r≤ n. We need to prove (n-r+1)nCr-1=n(n-1Cr-1)

Q: Find the chara derieti'k palynow ral of f the aatrcx 2. 1)

A: Answer is mentioned below

Q: b. Let x, y,p E Z, where p is prime. Prove that x2 - p²y - p + 0. (Hint: you may use without proof…

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Q: What are the value for sup e-* = е x>0 and inf e * - = XER

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Q: Then show that N(A) = R(AT)+ and N(AT) = R(A)+. %3D 11-1 02-3 -1 1-2 N(A) = N(AT) =

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Q: Let x, y,p E Z, where p is prime. Prove that x2 – p²y – p + 0. (Hint: you may use without proof that…

A: Solution :-

Q: 2 O Expand Pn = 2x +3x +) in Term of P.f,.

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Q: Suppose that X ~ Unif(-1,1). Compute (d) P(X = 0.5), (e) P(X² > 0.5).

A: Let ,  The pdf of X is ,  = 0 ; otherwise Our aim is to find the following :  d) P(X=0.5) e)…

Q: of 3 ty ?

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Q: Prove that (A')' = A for each A E Mmxn(F).

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Q: Let P= (p:p=6x-1, for some x EZ) and Q = {q:q=6y+5, for some y EZ}. Prove that P=Q

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