Let X1, . . . , Xn, . . . be iid random variables with common PDF f(x | θ) = e^−(x−θ) x > θ, 0 x ≤ θ. We wish to show that the MLE ^θn = X(1) is consistent.   (a) Why can you not directly apply the MLE theorem to obtain this conclusion? (b) Prove that ^θn is consistent by using the CDF for X(n) and the definition of convergence in probability. (c) Prove that ^θn is consistent by considering the bias and variance of the estimator.

Q: A random process X(t) is defined as X(1) = A̟ cos(2Tf,1)+A, sin(27f,1) where A, and A, are…

A: Given the random process X(t) as Xt=Accos2πfct+Assin2πfct

Q: Suppose that X and Y are independent random variables each with a x'(2) distribution. Find the…

A:

Q: Let X1, X2 denote two independent variables, each with a x^2(2) distribution.  Find the joint pdf of…

A:

Q: 1 V(X) for i) X ~ U{1,n}, ii) X~ Geometric (p), iii) X P

A: We have given that X be a random variable which follows 1) Uniform distribution, i.e. X∼U{1,n}2)…

Q: Let x1,x2,...x n be an iid sample with pdf f(x|0)=0X (® -1) Osxs1,0>0 Find the MLE of 0.

A:

Q: Let x be a random n-vector with mean mu and positive definite variancecovariance matrix Sigma.…

A: Let's break down the problem into two parts(a) To show that Sigma(-1/2) (x-mu) has a mean of 0 and a…

Q: 7 Let X1,...Xn be iid Normal( θ+ c, σ^2), where c and σ ^2 are known constants (i.e., E(Xi) = θ +…

A:

Q: 4 Assume the PDFS of the two independent random variables X and Y are, |1, 0<x<1 e, 0<y fx (x) = and…

A: Given the pdfs of independent random variables X and Y as fXx=1  , 0≤x≤10  , else  and   fYy=e-y  ,…

Q: Let X be a Gaussian random variable (0,1). Let M = e^(5*X) be a derived random variable. What is…

A: Let  X be a Gaussian random variable with mean 0 and variance 1. We know that its probability…

Q: Determine the expected value of g(x,y) = (X^3)+(Y^2)

A: Solution

Q: Let X, i=1,20 be identical independent Poisson RVs distributed with parameter i = 2. Calculate and…

A: Given that X1, X2, X3,.....,X20are identical Poisson random variables Parameter of all random…

Q: Q2\\ let X be a continuous random variable with pdf: 0 <x <1 1 < x < 2 Sa(x) = {} otherwise Find the…

A:

Q: 12. Let X be a normal random variable with mean 678 and standard deviation 8.85. Let Q₁ the…

A:

Q: Prove that the rv Y=(tan^-1(X1/X2)) and Z=(X1^2+X2^2)^0.5 are independent.   No distribution is…

A:

Q: (i) Briefly explain how Moment generating Function of a random variable, y may be used to generate…

A: As per give by the question, The moment generating function of a real valued that random variable is…

Q: Q4) If X is a continuous random variable having pdf ke~ (2x+3y) x>0y>0 xy) = = e p(x) { 0 otherwise…

A: Given data: To find:

Q: Let X be a Poisson random variable with parameter k. (a) Show that the moment generating function…

A: Here X~ Poi(k). Then, P(X=x)=e-kkxx!,x=0,1,2,...0,otherwise a.)The Moment Generating Function of X…

Q: Let (x₁, Y₂) (i = 1,...,n) be independent following the Poisson regression model (without intercept)…

A: Hi! Thank you for the question, As per the honor code, we are allowed to answer three sub-parts at a…

Q: Suppose that X is a continuous unknown all of whose values are between -3 and 3 and whose PDF,…

A: Find the constant c: The value of the constant c is obtained as 1/36 from the calculation given…

Q: Suppose the random variable X~ Beta(a, 3), namely its PDF fx(x) = I(a + B) r(a)r(3)* r^-1(1−z)ß-1, 0…

A: Solution

Q: The density of a random variable X is f(x) = C/x^2 when x ≥ 10 and 0 otherwise. Find P(X > 20).

A: Given:fx=cx2,                                   x≥10…

Q: The 4-dimensional random vector X has PDF fX(x)={1 when 0≤xi≤1, i=1,2,3,4 and 0 otherwise}. Are the…

A: Step 1: Step 2: Step 3: Step 4:

Q: the pdf of a random variable x is given by f(x;θ) = θx^θ−1 x ∈ (0,1). find the maximum likelihood…

A: The likelihood and log likelihood functions are as follows:

Q: The joint PDF of a two dimensional random variable (X, Y) is given by f(x. y) = xy + ,01) (ii) P(y…

A: Given:- f(x, y)=x y2+x28,0≤x≤2,0≤x≤1

Q: Let X be a random variable with p.d.f., (2e2(1-x),x>1 0 ,x<1. (1) Find the moment generating…

A:

Q: Let X ~ Geom(p). (i) Compute the moment generating function, and show your computation. Your answer…

A: We want to mgf, first and second moment and variance of the geometric distribution

Q: a) Calcuate the exact expectation and the variance of X.

A: Given: X = max(10,Y) with Y ~ Poisson λ=13

Q: Suppose X is a random variable with the cdf Fx(x) = 1-(1 + x²) x>0, C70, X>0 Derive the pdf of the…

A: Suppose X be the random variable with the cdf  FXx=1-1+xc-γ  ; x&gt;0, c&gt;0, γ&gt;0

Q: Suppose that two independent and identically distributed random variables X and Y have common mgf…

A: Two independent and identically distributed random variables X and Y have common mgf:- MX(t) = MY(t)…

Q: The discrete random variable T has the p.m.f   P (T=m)=2^m -1 / 3^m , m=0,1,...,m,... Show that…

A:

Q: Let X1,..,Xn be a random sample from Unif(0, 0+1). Let ô, = X - 62 = X(n) - %3D n+1 Find the…

A:

Q: For an exponential random variable (X) having θ = 4 and pdf given by: f(x) = (1/θ)e^(−x/θ ) where x…

A: We want to find the mean and variance and P(X&gt;3)= ?

Q: Suppose that two random variables X and Y have the joint PDFfXY(u, v) = {60(u^2)v…

A: Step 1:Step 2:Step 3:

Q: Let X be a random variable with moment generating function M(t), and let Y= ax+b. Prove that the…

A: Solution-: Let, X- be the random variable with MGF M(t) Let, Y=ax+b We prove that, M(t)=ebt*M(at)

Q: If X is a continuous random variable with X ∼ Uniform([0, 2]), what is E[X^3]?

A: is a continuous random variable following a uniform distribution, that is .

Q: ew] (i) Find the mean and variance of X using the pdf. (ii) Find the moment generating function of…

A: Given the pdf: f(x) = { (1/6)e^(1/6x) ; x &gt; 00; ew]

Q: Find auto-correlation function of a random process whose power spectral density is given by 1+

A:

Q: a) You throw two fair dice labeled 1, . . . , 6 independently of each other and let X be the minimum…

A: a) The possible out comes of rolling a pair of die are, Let 'x' be the minimum of the ordered pair.…

Q: Find auto-correlation function of a random process whose power spectral densis is given by 1+ 4

A:

Q: Let random variables X and Y have the joint pdf   fX,Y (x, y) = 4xy, 0 < x < 1, 0 < y < 1   0,…

A: Let X and Y be the continuous random variables  with the following joint densityThe defined new 2…

Q: Let X be a continuous random variable with a pdf given by f(x) = (3x^2)/2 where x lies in the closed…

A:

Q: Consider the data (1.3). (2.5). (3

A: Given the data          1, 3, 2, 5, and  3, 9.67

Q: Find a complete sufficient statistic.

A: