Prove the following Theorem:Let X₁,..., X₂, be iid observations from a pdf or pmf f(z) that belongs to an exponentialfamily given by(Źw.(09)t.(z)),£4(X))f(x)=h(x)c(@)expw)t(xwhere = (01,...,04), d≤k. ThenT(x) = (24(x;),-j-1is a sufficient statistic for.

The probability density function is given as f(x)=h(x)c(θ) exp({∑i=1kwi(θ)ti(x)Using the…

Answered: Prove the following Theorem: Let…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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3. Let X; N iid U(0, 1) for i e {1, 2, ...,n}. (a) Compute the pdf of Y; = - log(X;), name the distribution that Y; follows and provide any parameter value or values needed. (b) Compute the pdf that W = [I", X, = X1 X2... X, follows. You may use the result given as part of the Example 2.6 video (even though it will only be proved in Chapter 3). (c) Compute E[W] in the two following ways: (i) through exploiting independence of the X, and using EUV] = E[U]E[V] for any independent random variables U and V. (ii) using the pdf of W computed in the previous part.
Find auto-correlation function of a random process whose power spectral densis is given by 1+. 4
Let X1,..., Xn be a random sample from a uniform distribution on the interval [20, 0], where 0 0. Let X(1) < X(2) <...< X(n) be the order statistics of X1, ..., Xn.
der a random sample X1, X2, ..., X, having the pdf, f(r; 0) 1
Four data points, (0,1), (1,4), (2,2), and (3,3), were collected in an experiment. The DFT of [1, 4, 2, 3] is [5, - - i, -2, - +zi]. Let f (t) + 늑1. Let f (t) = a + b cos( nt) + c sin(÷ Tt) + d cos(at) be the trig interpolation function that passes through the four data points. What is the value of the coefficient d?
Page 263, 4.4.8
Find the instantaneous form for each of the following phasors: 20 (a) A, = j10a, +a (b) B, = j4e-*a, + 6e*P*a, (c) C, = j2e=20%e¬J#lda -20z
Which of the following is the correct sum of squared residuals to be minimized to find tne fit f(x) = £ (2r-1)"* - using a data set ((xo, Yo), (x1,y1), --, CKn. Yn)} ? A.) Eo(In(y;) + a – b2x; + 1)² B.) E-(In(y;) + In a + b In(2x; – 1))² D.) E7-o(In(y;) – In a – b In(2x; – 1))² E.) E-o(In(y;) + In a – b2x; + 1)²
2) Let X₁, X2, ..., Xn be a random sample from the pdf f(x;0) = 0xª−¹, 0≤x≤ 1,0 <0 < ∞. Find the MLE of 0 and show that its variance approaches 0 as n approaches ∞o.
2.21 Let the random variable X have pdf { f(x) = = 49 X - 3 0 < x < 1, 1 < x < 2. Generate a random variable from f(x), using a) the inverse-transform method, b) the acceptance-rejection method, using the proposal density g(x) = 1/2/₁ 0 ≤ x ≤ 2.
Suppose that X is a continuous unknown all of whose values are between -5 and 5 and whose PDF, denoted f, is given by f ( x ) = c ( 25 − x^2 ) , − 5 ≤ x ≤ 5 , and where c is a positive normalizing constant. What is the expected value of X^2?
page 328 5.2.2

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Prove the following Theorem: Let X₁,..., X, be iid observations from a pdf or pmf f(z) that belongs to an exponential family given by S({z}8) — A(z)c(@?exp (Ém(1)t,(a)). f(x Σ TUX)-(XX)) where @ = (0,,04), d ≤ k. Then is a sufficient statistic for 6.