3. Suppose that ai . ,,an are a random sample from a N( ,02) distribution. Recall...
We have a random sample of size 17 from the normal distribution N(u,02) where u and o2 are unknown. The sample mean and variance are x = 4.7 and s2 = 5.76 (a) Compute an exact 95% confidence interval for the population mean u (b) Compute an approximate (i.e. using a normal approximation) 95% confidence interval for the population mean u (c) Compare your answers from part a and b. (d) Compute an exact 95% confidence interval for the population...
Recall that ?-n/ ??-1 log Xi is the mle of ? for a beta(8.1) distribution.Also -_ ? ial log Xi has the gamma distribution ?(n.18) (a) Show that 2eW has a x2 (2n) distribution (b) Using part (a), find c1 and c2 so that 2?? for 0 < ? < 1 . Next, obtain a (1-?)100% confidence interval for ? (c) For ? = 0.05 and n = 10, compare the length of this interval with the lengthof the interval...
6.2.1 2. Recall that θ--r/ Σ (θ, 1 ) distribution. Also, W - i-1 log Xi has the gamma distribution Г(n, 1/ ) -1 log X, is the mle of θ for a beta (a) Show that 2θW has a X2(2n) distribution. (b) Using part (a), find ci and c2 so that (6.2.35) for 0 < α < 1 . Next, obtain a (1-a) 100% confidence interval for θ.
Let X1, X2, ..., Xn be a random sample from the N(u, 02) distribution. Derive a 100(1-a)% confidence interval for o2 based on the sample variance S2. Leave your answer in terms of chi-squared critical values. (Hint: We will show in class that, for this normal sample, (n − 1)S2/02 ~ x?(n − 1).)
. Let Yı, . . . , Ý, be a sample from N(0, σ*) distribution. Show that both Gi (Yi, . . . , X,; σ) = nHare pivots. j-1 72 and G (1) Recall the confidence interval based on Gi that we derived in class. (2) Let Z be N(0, 1) random variable. Find the expectation and variance of |Z. (3) If n is large, what is the approzimate distribution of (4) Use (3) to construct an approximate confidence...
. Let Yi, ,Ý, be a sample from N(μ, σ2) distribution, where both μ and σ2 are un known Repeat the argument that was given in class to show that is a pivot (start by representing Yj as a linear function of a N(0, 1) random variable). Use the fact that (n-pe, of freedom") to construct the confidence interval with coverage probability 95% for σ2 (you can state the answer in terms of quantiles of X2-distribution, or find their numerical...
em 3. Let Xi. A.2. . . . A., be i. i.d. random variables from an exponential diatribatnn-nsmesn be i.i.d. random variables from an exponential distribution with mean Ame and let } samples are independent. Recall that an exponetial random variable with mesn 9 hiss deaity 0 (a) Assuming that θ = θ-θ2, find the MLE of θ when X!, . . , Xn and Yi, ,Yn are observed. (b) Find the LRT to test the hypothesis that θ,-, versus...
1. Suppose you are drawing a random sample of size n > 0 from N(μ, σ2) where σ > 0 is known. Decide if the following statements are true or false and explain your reasoning. Assume our 95% confidence procedure is (X- 1.96X+1.96 Vn a. If (3.2, 5.1) is a 95% CI from a particular random sample, then there is a 95% chance that μ is in this interval. b. If (3.2.5.1) is a 95% CI from a particular random...
Suppose you have selected a random sample of n=9 measurements from a normal distribution. Compare the standard normal zz= values with the corresponding t values if you were forming the following confidence intervals. (a) 98% confidence interval z= t= (b) 95% confidence interval z= t= (c) 99% confidence interval z= t=
1.Suppose X1, X2, .., Xn is a random sample from N(", 02) 10 pts] If o2 1, u is unknown. Find the MLE of a. b. [10 pts If o2 = 1, p is unknown. f = X is an estimator of u. What is the MSE of this estimator? Now assume o2 is unknown. The following data is a set of observations of X1,..., Xn. Use the dataset to answer (c), (d) and (e) 11 8 9 7 6...