Under the SLR model, Y₁, Y2, . . ., Y, are independent normal random variableswith E[Y;] = ẞo + ẞ₁x; and V(Y;) = σ², for i = 1, 2, ..., n.a. Compute the likelihood function, L(y1, 2, ..., Yn Bo, ß1,0²) = L(ẞ0, B1, 0²).b. Show that the MLE of o² is SSE [the MLEs ofnBo and B₁ are the same as the least-squares estimators, Bo and B₁.]

a. Likelihood Function in SLR modelThe likelihood function in a Simple Linear Regression (SLR) model…

Answered: er the SLR model, Y₁, Y2, ... Yn are…

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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Let X1, X2, . . . , Xn be an i.i.d. random sample from a Beta distribution with density: f(x; θ) = Γ(2θ) Γ(θ) 2 x θ−1 (1 − x) θ−1 , 0 < x < 1, θ > 0. Find a sufficient statistic
Consider a random variable Y with PDF Pr(Y=k)=pq^(k-1),k=1,2,3,4,5....compute for E(2Y)
Consider a real random variable X with zero mean and variance σ2X . Suppose that wecannot directly observe X, but instead we can observe Yt := X + Wt, t ∈ [0, T ], where T > 0 and{Wt : t ∈ R} is a WSS process with zero mean and correlation function RW , uncorrelated with X.Further suppose that we use the following linear estimator to estimate X based on {Yt : t ∈ [0, T ]}:ˆXT =Z T0h(T − θ)Yθ dθ,i.e., we pass the process {Yt} through a causal LTI filter with impulse response h and sample theoutput at time T . We wish to design h to minimize the mean-squared error of the estimate.a. Use the orthogonality principle to write down a necessary and sufficient condition for theoptimal h. (The condition involves h, T , X, {Yt : t ∈ [0, T ]}, ˆXT , etc.)b. Use part a to derive a condition involving the optimal h that has the following form: for allτ ∈ [0, T ],a =Z T0h(θ)(b + c(τ − θ)) dθ,where a and b are constants and c is some function. (You must find a, b, and c in terms ofthe information…
If X and Y have the joint probability distributionf(−1, 0) = 0, f(−1, 1) = 1 4 , f(0, 0) = 16 , f(0, 1) = 0, f(1, 0) = 112 , and f(1, 1) = 12 , show that (a) cov(X, Y) = 0;(b) the two random variables are not independent.
Let X1,...,Xn be iid random variables with expected value 0, variance 1, and covariance Cov [Xi,Xj] = ρ, for i≠j. Use Theorem of linearity of expectation to find the expected value and variance of the sum Y = X1 +...+Xn.
(b) Let Z be a discrete random variable with E(Z) = 0. Does it necessarily follow that E(Z³) = 0? If yes, give a proof; if no, give a counterexample.
consider x-U(0,1). Let Y=eX and Z=X2. Find the expected value of random variable T=Y+Z
Find the variance by calculating the first two moments of the random variable X = (- 1 / λ) ln (1-U), where U ~ U (0,1) and λ> 0.
If X is a continuous random variable with X ∼ Uniform([0, 2]), what is E[X^3]?
Let X1,...,Xn be iid exponential(θ) random variables. Derive the LRT of H0 : θ = θ0 versus Ha : θ 6= θ0. Determine an approximate critical value for a size-α test using the large sample approximation.
Let X1 and X2 be two independent random variables. Suppose each Xi is exponentially distributed with parameter λi. Let Y=Min (X1, X2). A) Find the pdf of Y. B) Find E(Y). Hint: Let Y = Min (X1, X2). 1. P[Y > c] = P[Min (X1, X2) > c] = P[X1 > c, X2 > c] 2. Obtain the pdf of Y by differentiating its cdf of Y.

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