Let X₁, X2,..., Xn be a random sample on the random variable that is uniformly distributed over the interval (1,0) Cit 1:1
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- Suppose that the random variables X1,...,Xn form a random sample of size n from the uniform distribution on the interval [0, 1]. Let Y1 = min{X1,. . .,Xn}, and let Yn = max{X1,...,Xn}. Find E(Y1) and E(Yn).Let X₁,X₂,...,Xₙ denote a random sample from a distribution that is N(0,θ), where the variance θ is an unknown positive number. Show that there exists a uniformly most powerful test of size α for testing the simple hypothesis H₀ : θ = θ', where θ' is a fixed positive number.Suppose that the random variable X is continuous and takes its values uniformly over the interval from 0 to 2. What is P{X = 1.5 or X = 0.4}?
- Suppose that three random variables X1, X2, X3 form a random sample from the uniform distribution on interval [0, 1]. Determine the value of E[(X1-2X2+X3)2](a) Let X1, X2, X3, X4 be IID uniform on (0,1) random variables and let S4 = X1 +X2 +X3 +X4. Use normal approximation to calculate P(S4 ≥ 3) approximately by linear interpolation. b)Suppose the IQ scores of a million individuals have a mean of 100 and SD of 10. Without making any further assumptions about the distribution of the scores, find an upper bound on the number of scores exceeding 130.Let X1...., Xn be a random sample of size n from an infinite population and assume X1 d= a + bU2 with the constants a > 0 and b > 0 unknown and U a standard uniform distributed random variable given by FU (x) := P(U ≤ x) = 0 if x ≤ 0 x if 0 < x < 1 1 if x ≥ 1 1. Compute the cdf of the random variable X1. 2. Compute E(X1) and V ar(X1). 3. Give the method of moments estimators of the unknown parameters a and b. Explain how you construct these estimators!
- Let X and Y be independent Poisson random variables with means 2 and 8, respectively. More explicitly, their probability mass functions are given by Px (k)=e¹ PY (k)= e and let Z=X+Y Find the variance of X given that Z=10, that is Var(X|Z=10). Express the result with two decimal points, Ex 12.34 Yaret: for k=0,1,... for k=0, 1,...Let X1, X2, ... Xn random variables be independent random variables with a Poisson distribution whose parameters are l1, l2, ... ln, respectively. Which of the following is the moment generating function of the random variable Z defined as (the little image)?Consider a random sample X1,...,Xn,... ∼ iid Beta(θ,1) for n > 2. Prove that the MLE and UMVUE are both consistent estimators for θI got MLE = n/-∑logXi and UMVUE = (n-1)/∑logXi. Need help in proving consistency
- Let X1, ..., Xn be a random sample from N(μ, σ2), where σ2is known.a) Show that Y = (X1 + X2)/2 is an unbiased estimator of μ.b) Find the Cramer-Rao lower bound for the variance of an unbiasedestimator of μ for a general n.c) What is the efficiency of Y in part (a) above?Let X1, X2, ... , Xn be a random sample from N(μ, σ2). Find the Moment Generating Function of X̅. If n = 16 and σ = 2, compute P(-1 ≤ X̅ - μ ≤ 1).Suppose that X1, . . . , Xn form a random sample from the uniform distribution on the interval [θ1, θ2], where both θ1 and θ2 are unknown (−∞ < θ1 < θ2 < ∞). Find the MOM estimates of θ1 and θ2.