Let X ~ Geom(p), a geometric distribution with parameter p € (0, 1). This is a discreteprobability distribution on the positive integers, with p.m.f. given byfx (k) = (1 − p)k-¹p, k == 1, 2,...(i) Verify (by calculation) that the log-likelihood function for the parameter p is given byl(x;p) = n (logp + (x − 1) log(1 − p)).-(ii) Derive the MLE estimator for p, and use it to compute the MLE estimate for thefollowing sample data ¹:1 3 1 1 1 2 3 1 1 1 1(iii) What is the MLE estimator for 1/p? Verify this estimator is unbiased.

There is a random variable , that follows the geometric distribution with the parameter .That is;…

Answered: (i) Verify (by calculation) that the…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Consider the following data on price ($) and the overall score for six stereo headphones tested by a certain magazine. The overall score is based on sound quality and effectiveness of ambient noise reduction. Scores range from 0 (lowest) to 100 (highest). Brand Price ($) Score 180 74 В 150 69 95 63 70 56 70 40 F 35 28 (a) The estimated regression equation for this data is ý = 25.870 + 0.291x, where x = price ($) and y = overall score. Does the t test indicate a significant relationship between price and the overall score? Use a = 0.05. State the null and alternative hypotheses. O Ho: Bo * 0 H: Bo = 0 O Ho: B1 20 Hoi Bq < 0 O Ho: Bo = 0 Hg: Bo 0 O Ho: B1* 0 H: B, = 0 O Ho: B, = 0 H: B1 = 0 Find the value of the test statistic. (Round your answer to three decimal places.) Find the p-value. (Round your answer to four decimal places.) p-value = What is your conclusion? O Do not reject H. We conclude that the relationship between price ($) and overall score is significant. O Do not reject…
Test the claim about the mean of sample 1, µ1 has a lower than the mean of sample2, µz at the level of significance a. Assume the samples are random and independent, and the population are normally distributed. Claim: 41 = µ2; a = 0.01 Assume the population variances are not equal Given: sample 1: ī = 3037 s = 706.3 n = 205 sample 2: = 3272 s = 660.2 n = 195 • The hypothesis test is Ho: µ1 = µ2 H1: Pi Answer choose one of these: Type A for =; Type B for and Type D for + Test statistic:t = (Two decimal places.) P- value is (four decimal places.) Construct confidence interval at a = 0.10 < µi - H2 <
Assume that both populations are normally distributed. (a) Test whether µ, + µ, at the a = 0.01 level of significance for the given sample data. Population 1 Population 2 n 11 11 (b) Construct a 99% confidence interval about µ, - H2. 18.8 26.7 S 3.8 5.3 (a) Test whether µ, # µ2 at the a = 0.01 level of significance for the given sample data. Determine the null and alternative hypothesis for this test. A. Ho:H1 = H2 B. Ho:H1 = H2 H1:H1> H2 Ho:H1 # H2 H1:H1 = H2 D. Ho:H1 # H2 Detemine the P-value for this hypothesis test. P= (Round to three decimal places as needed.)
Assume that you have a sample of n, = 8, with the sample mean X, = 47, and a sample standard deviation of S, = 7, and you have an independent sample of n, = 7 from another population with a sample mean of X, = 37 and the sample standard deviation S, = 8. Assuming the population variances are equal, at the 0.01 level of significance, is there evidence that u, > H2? Determine the hypotheses. Choose the correct answer below. O A. Ho: H1 SH2 H1: Hy> H2 O B. Ho: H1 =H2 H: H1 # H2 O D. Ho: H1 H2 O C. Ho: H1> H2 H1: H1 SH2
6
(Mathematical Statistics) Let X1,X2,...,X5 be a random sample of the Gamma distribution with parameters a = and ß = 0, e > 0. Determine the 95% confidence interval for the parameter e based on the statistic E(i=1 - 5) Xi. 2
q5-
(CLT approximation) Suppose that X is a r.v. with mean 6 and variance 10, and X1,...,X100 are i.i.d. copies of X. (a) Use the Central Limit Theorem to approximate P (?100 Xi ≥ 1000). i=1 (b) Suppose that ?ni=1 Xi = 28. Use the Central Limit Theorem to compute (1 − α) approximate confidence intervals for α = 0.05.
ANOVA table is presented as: (a) (b) Source of Variation Factor A (Capping material) Factor B (Batch) Interaction (AB) Error (E) Total (T) SS 11573.38 17930.09 1734.17 4716.67 35954.31 Test all hypotheses from the ANOVA table using a 0.05 level of significance. Perform a pairwise comparison test using Fisher's LSD procedure using a 0.01 level of significance.
Mental measurements of young children are often made by giving them blocks and telling them tobuild a tower as tall as possible. One experiment of block building was repeated a month later, with the times (in seconds) and five pairs of children’ completion times are listed below. Test the claim that there is a difference in times between two trials. (a) At α = 5%, use the Classical approach to conclude it. (b) Using the P-value approach to conclude it. 1st Trial (A) 33 28 29 26 29 2nd Trial (B)30 25 24 24 29.
(b) Let μ1 be the population mean for x1 and let μ2 be the population mean for x2. Find a 95% confidence interval for μ1 − μ2. (Round your answers to two decimal places.) lower limit % upper limit %

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