Suppose X~exp(λ), Use Neymann-Pearson Lemma, Most Powerful Test, and Likelihood ratio test to test hypothesis Ho:λ=1 vs Ha:λ=4. Assume ɑ=0.05
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Suppose X~exp(λ), Use Neymann-Pearson Lemma, Most Powerful Test, and Likelihood ratio test to test hypothesis Ho:λ=1 vs Ha:λ=4. Assume ɑ=0.05
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- Suppose X~ Binom (6,p) and define an estimator T=X/6 to estimate p. Then variance of T isLet X be a random variable with mean µ and variance σ^2. Let Y = aX2 + bX + c. Find the expected value of Y.Show that for a random variable X with mean μ and variance σ^2, the standardized random variable Z corresponding to X has mean 0 and variance equals 1 by using the properties of expectation and variance.
- Show that the variance of a geometric random variable X; is given by V ar(X) = q/p^2 .A researcher that wanted to estimate the expectation AY of a random variable Y got three independent observations, Y, Y, Y The researcher knows the value o, of the variance of Y and is considering the following estimators: Pi = 4) Yi + (+) ¥ Py = () Yn + (;) ¥½ + () Y½ in (}) Yi + (}) ¥z + (() %D Which of the following is correct? ONone of the above Ois an unbiased estimator of l and it has the smallest variance of the three estimators. Ois an unbiased estimator of µ and it has the smallest variance of the three estimators. O and i, are both unbiased estimators of fl and Var (ſîz) < Var (îì3). is an unbiased estimator of µ and it has the smallest variance of the three estimators.Suppose that f (x) = e=* for 0 < x Determine the mean and variance of the random variable.
- Suppose that X and Y are random variables where the conditional expected value of Y given X is given by E (Y|X) = 3X + 6 If X has the expected value 0 and variance Var(X)=9 determine the minimal possible value of the variance Var(Y) of Y./Suppose X1, X2, X3, X4 be i.i.d. normal random variables with mean 0 and variance sigma^2, where sigma^2 is the unknown parameter. Consider the following estimators: T1 = X1 - X2 + X4, T2 = 1/3(X1 + X1 + X4), T3... T4... T5 = 1/2|X1 - X2| (a) Is T1 unbiased for sigma^2, for i = 1,2,3,4 (b) Among the estimators T1,...,T4 for sigma^2, which has the smallest MSE? (c) Is T5 unbiased for sigma? If not, find a constant k so that k*T5 is unbiased for sigma^2. Evaluate the MSE of T5.Suppose X1, X2,...,X, from a real-valued random variable X with unknown mean u and standard deviation o. What is the estimator of the signal to noise ration u/o.?
- If null hypothesis is false because a drop has occurred, then the observed z will be?Derive the variance of the Student's t distribution using the definition Var (x) = ELX2] - EX)² .in a hypothesis test for the difference of two proportions, the point estimate is calculated to be ?̂ 1−?̂ 2=0.11p^1−p^2=0.11, and the standard error is calculated to be ??=0.0099SE=0.0099. Compute the test statistic if the null hypothesis is that the two proportions are equal.