iid 1. Let Y1,..., Yn * Bernoulli(p). It can be shown using the properties of the MLE that when n is sufficiently large: yn(Ỹ – p) _d, N (0, 1) VÝ(1 – Ý) (a) Use this result to show that: - p) S Za/2 VÝ(1 P 21– a Y) (b) Use your answer in the previous part to derive an approximate 1 – a confidence interval for p.

Q: 1 zat 1 + 22

A:

Q: . Find the Z transform of : x(k) = (2)*sin,k

A: Z transform

Q: dr Evaluate f x In(x) In(In(x))

A: Evaluate: ∫dxxlnxlnlnx.  Let lnlnx=u. Derivate with respect to x by applying chain rule:…

Q: (b) En=1 m

A:

Q: 18. Let T) Σ-1() k=0 (a) Use Pascal's Triangle to compute T(n) for n = 1, 2, 3, 4. (b) Prove that…

A: We have

Q: TheZ- Transform. Show. that. xe ..Z X(2)_ assumeera 1.C.

A:

Q: 2. Find all volves of tand u x(t,4)=0 and simultomebusly such that =0. 3. 5x-11* C.

A: Get the equations for:  fx=0fy=0 15x2-11y=0-11x+3y2=0 or 15x2=11y3y2=11x

Q: P(Z'UY') =

A:

Q: 2) Recall F(X = t) = Pr(X < t) calculate the following using the Z transform and F(Z=t) for the…

A: According to our guidelines we are solving only first three sub-parts of a question. Given that F(X…

Q: Aagamant # os Q#os Find he harploce truriJerm og the Jenetim EBinut 26 Ginan that 7(2125) = 7(t)

A: Laplace transform of periodic function: The Laplace transform of a periodic function f(t) of period…

Q: X ∼ Poisson(θ) and independently, Y ∼ binomial(2, θ). Z = XY. (c) Define W = Z−(1+2θ−θ2)X. Are Z…

A: X~Poisson()Y~bnomial(2,)Z = XY

Q: . Let Dn be as in Example 10. Find a Hamiltonian circuit inCay({(r, 0), (f, 0), (e, 1)}:D4 ⨁…

A: Introduction: A Hamiltonian circuit is a circuit in a graph that visits every vertex exactly once…

Q: Q2) Discus the existence and uniqueness of the following IVPS: y = x + ys y = x tan (i) (ii) y(1) =…

A: We Know that,

Q: 3. Find dz/Ou and dz/du. (a) z = 9x*y- r+ 5y; X = UV; Y =v – u (b) z = 2e*". 1 a= vuv: y=

A:

Q: Let P(Z)=0.43, P(Y) = 0.42, and P(ZnY) = 0.18. Use a Venn diagram to find (a) P(Z'nY'), (b) P…

A:

Q: Let (An) and (Bn) be (Fn)-martingales on (2, F, P) and let Cn = A - Bn. Show that E [(Am - An) |Fn]…

A: Solution:

Q: Find the Fourier Transform of the triangular pulse x(t) = { (1– t for -1<t<0 for (1 +t 0<t <1

A:

Q: Sirlw)dk= f (x) κ= lim Σ fx%) Δx 25.) Use the definition, -2x n+00 i=1 to evaluate Σ Ei=" (n+1) Eř="…

A:

Q: Let X be a random variable such that P(X= c+kh) = π(a) (poisson distribution), where c s a real and…

A: In this problem, we are given a random variable X that follows a Poisson distribution with parameter…

Q: 9- Let X-U(0,0), then P(X> X²) is (a) 0 (b) 1/0 (c) 1/2

A: Given that, X~U(0,θ)

Q: 11. The region D bounded by x ≥ 0, z ≥ 0, y ≥ 3x and y² + 2² ≤ 9 is shown in the figure. JSS z dV =…

A: The given region is: x≥0,   z≥0,  y≥3xy2+z2≤9

Q: Researchers at a drug company are testing the duration of a new pain reliever. The drug is normally…

A:

Q: =)²dx<∞. Let U₁, U2,... be IID Unif(0, 1) random variables. F 1 In = Σƒ(U). i=1 Argue that µ =…

A:

Q: Q.4) The Wronskian W(e-x, e*, e2x), is: a) 6e% Q.5) The general solution of the equation yto + 9y" =…

A:

Q: Find the I-coordinate e a Hhe ocal monimeno 07 local makimum of 7 qunction n 7ux)=e? (2x²+3x), x ER

A: Second derivative test : Suppose f is continuous  and differentiable function at x=a where f'a=0 and…

Q: Let Dn = {ar , a; b | 0<k< n}, where a " = e, b² = e , and ba =alb (a) Show that (a* b)? = e for…

A:

Q: (d)) A random process is defined as X(t) = µ + N(t), where N(t) is white noise with auto covariance…

A: Given: A random process is defined as                                         , where  is white…

Q: find Fourier's 0 f(x) = 2 -1≤X20 06X≤1

A:

Q: Q2) If y(t) is the convolution between x(t) = 3 rect () and h(t) = rect () then find Ten of the…

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Q: Compute the Wronskian W(n-92)(t) for t >0 where (t) = t int, 2(t)=2. A. nt+1) B. Pat-1) C.fat D.…

A:

Q: The poaching model for Species Z is Z' = 0.000923 Z(1- Z) -0.002 where the variable Z represents the…

A:

Q: Determine DFT X[ej"] of the signal (answer a or b) a) x[n]coswon. b) x(n) = a"u(-n- 1) %3D

A: As it's mentioned in the question to answer between a) or b), so showing steps for only one part. b)…

Q: A cubic spline S is defined by: (So(x)%D 1+x- x (5,(x) = 2 - 5(x – 1) + (x – 2) if 0 SxS1 if 15xS2…

A: As we all know that Natural Cubic Spline is a piece-wise cubic polynomial that is twice continuously…

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Q: Which of the following is a nonlinear second order nonhomogenous PDE a. xux-Uxx=tuxt O b.…

A: To find - Which of the following is a nonlinear second order non-homogenous PDE x ux - uxx = t uxt…

Q: Suppose that X1,..., Xn are i.i.d. Poisson(0) where 0 > O is an unknown parameter. The prior T(0) on…

A:

Q: R and r are defined by E(r, – 5)(y – T) Sry VS., Syy SSR R2 r = SST' (1) Show R? = r².

A: If y = a+bx is the linear equation  Correlation attempts to measure the strength of the relationship…

Q: Let z = x + yi, z, = 1- 0.5i, and z, + z, = 3-1.5i. Find In(z,) - In(z2) C) In(50) + 2nin A) In(25)…

A:

Q: The Black-Scholes equation provides a model for certain types of transac- tions in financial…

A: In the given qurstion we have to explain the equation which writen under blue sign we will explain…