Show that (X+1)/(n+2) is a biased estimator of the binomial parameter θ. Is this estimator asymptotically unbiased
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Show that (X+1)/(n+2) is a biased estimator of the binomial parameter θ. Is this estimator asymptotically unbiased?
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- Repeat Example 5 when microphone A receives the sound 4 seconds before microphone B.Respiratory Rate Researchers have found that the 95 th percentile the value at which 95% of the data are at or below for respiratory rates in breath per minute during the first 3 years of infancy are given by y=101.82411-0.0125995x+0.00013401x2 for awake infants and y=101.72858-0.0139928x+0.00017646x2 for sleeping infants, where x is the age in months. Source: Pediatrics. a. What is the domain for each function? b. For each respiratory rate, is the rate decreasing or increasing over the first 3 years of life? Hint: Is the graph of the quadratic in the exponent opening upward or downward? Where is the vertex? c. Verify your answer to part b using a graphing calculator. d. For a 1- year-old infant in the 95 th percentile, how much higher is the walking respiratory rate then the sleeping respiratory rate? e. f.Assume that the entire sample has 2 positive observations and 6 negatives observations. Variable X1: at the left branch has 2 positive observations and 4 negatives observations;at the right branch has 6 positive observations and 6 negatives observations. What is the information gain or reduction in uncertainty of X1 using the Gini index? (round to two decimal spaces)
- Suppose that a sequence of mutually independent and identically distributed discrete random variables X₁, X₂, X3,..., X₁ has the following probability density function (exe-e x! 0, f(x; 0) = for x = 0,1,2,... elsewhereShow that the mean of a random sample of size n from an exponential population is a minimum variance unbi-ased estimator of the parameter θ.Suppose the conditional mean function is Y = Bo + BiX + B2X² + B3X3 + U %3D where E[U]X]3D0. By mistake, a researcher omitted X^2 and X^3 terms in the regression and ran regression of Y on X and an intercept only. It turns out that covariance between X and B2X2 + B3X° is nonzero. Is the OLS estimator researcher computed consistent for beta 1? Yes, since the omitted variables are just functions of X. Yes, since the covariance between X and B2X2 + B3X is nonzero No, since the covariance between X and B2X4 + B3X° is nonzero No, since the omitted variables are negligibly small.
- Derive that the variance of the forecasted error using Exponential Smoothing (ES) method is given by 20² Var(et) 2 – α Where o² is the variance of individual observation and a is the parameter in the ES method. =Suppose that @, and ô, are unbiased estimators of the parameter 0 and that V@,) = 15 and V(@2) = 4. What is the relative efficiency of the two estimators?If X₁, X2,..., Xn constitute a random sample of size n from an exponential population, show that X is a sufficient estimator of the parameter 0.
- Suppose that the random change in value of a financial asset is X over the first day and Y over the second. Suppose also that Var(X) =18 and Var(Y) = 26 In this case, the total change in the value over these two days is given by X +Y. Do you have enough information to compute Var(X +Y)? If so, compute this value. If not, explain what additional information you need to do so.1-X = ¹/2 (X₁ + · · · + Xn) is an unbiased estimator for the population mean. FALSE 2-S² = ¹₁((X₁ - Ă)² + TRUE + (Xn - X)²) is an unbiased estimator for the population variance. TRUE FALSE 3-S = √S is an unbiased estimator for the population standard deviation. 6-If X₁, TRUE 4-If X₁,..., Xn are iid uniform (0, 0), with unknown, then 2X is an unbiased estimator for 0. TRUE FALSE 5-If X₁,. ‚ Xñ are iid N (0, 1), then X is also normally distributed. TRUE FALSE FALSE ‚‚ Xn are iid N (0, 1), then X is also normally distributed, with variance 1. TRUE TRUE FALSE 7-If X₁,..., X100 are iid N(0, 1), then the chance the 95% confidence interval (X - .196, X + .196) contains is .95. FALSE 8-Construct 100 confidence intervals for a population mean. The sample size is large, and use 95% confidence. About 5 of the intervals do not contain the population mean. TRUE FALSE 9-For X Binom(1000, p), the value of p that makes the variance of X largest is p = 0.5. TRUE FALSEQ3: Let X1. ,X, Binomail(1,0). (i) Show that B(1,0) is a member of the exponential class. (ii) Find the minimum variance unbiased estimator (MVUE) of 0.