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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A sample of n = 64 scores is selected from a population with µ = 80 and with σ = 24. On average, how much error is expected between the sample mean and the population mean?
A sample of n = 16 scores is selected from a population with μ = 100 and σ = 32. If the sample mean is M = 104, what is the z-score for this sample mean?
The amount of nicotine in a cigarette produced by a tobacco company is a random variable with mean 2.2 mg and standard deviation 0.3 mg. Taking 100 randomly chosen cigarettes, let X; denote the nicotine content of the ith cigarette for i = 1,..., 100, and let X = 100 Ei=1 X; be the sample mean of the nicotine content. 100 a. What is the variance of X? Var(X) = Approximate the probability that X is higher than 2.25 in terms of the standard normal random variable. (The answer is given in terms of 2.25, standardized by mean and standard deviation of X.) P(X> 2.25) = P(Z >
A sample mean, sample standard deviation, and sample size are given. Use the one-mean t-test to perform the required hypothesis test about the mean, μ, of the population from which the sample was drawn. Use the critical-value approach. Sample mean = 7.1 s = 2.3 n = 18 α = 0.01 H0: µ = 10 H1: µ < 10 The critical value(s) is/are (If there are two critical values separate each with a comma and list from smallest to largest)
9.5 Standard error of the mean. A random sample from a certain population has I = -44.6, s = 6.02, and n = 25. It is planned to conduct a test of Ha : µ << -40. (a) What is the null hypothesis in this case? (b) Do the summary statistics give some evidence that Ha is true? (c) Compute the estimated standard error of the mean, SEM= s/Vn. (d) Would it be surprising if Ho were true? Answer "yes" or "no," using two SEMS as the threshold of surprise. 35
A sample of n = 16 scores is obtained from a population with μ = 70 and σ = 20. If the sample mean is M = 80, what is the z -score for the sample mean?
The acceptable level for insect filth in a certain food item is 2insect fragments (larvae, eggs, body parts, and so on) per 10 grams. A simple random sample of 50 ten-gram portions of the food item is obtained and results in a sample mean of x=2.5 insect fragments per ten-gram portion. Complete parts (a) through (c) below.
Let x = age in years of a rural Quebec woman at the time of her first marriage. In the year 1941, the population variance of x was approximately ?2 = 5.1. Suppose a recent study of age at first marriage for a random sample of 41 women in rural Quebec gave a sample variance s2 = 2.9. Use a 5% level of significance to test the claim that the current variance is less than 5.1. Find the value of the chi-square statistic for the sample. (Round your answer to two decimal places.) What are the degrees of freedom?
Cigarette Smoking A researcher found that a cigarette smoker smokes on average 30 cigarettes a day. She feels that this average is too low. She selected a random sample of 9 smokers and found that the mean number of cigarettes they smoked per day was 32 . The sample standard deviation was 2.4. At α=0.10, is there enough evidence to support her claim? Assume that the population is approximately normally distributed. Use the critical value method and tables. (a)State the hypotheses and identify the claim. H0: claim/ do not claimH1: claim/do not claim This hypothesis test one tailed/two tailed a (Choose one) test. (b)Find the critical value. Round the answer to at least two decimal places. Critical value is c)Compute the test value. Round the sample standard deviation and score value to at least two decimal places as needed. F= P= (d)Make the decision. Reject/ do not reject (Choose one) the null hypothesis (e)Summarize the results. There ▼(Choose one) enough evidence to support…
Let m denote margin of error, n sample size and σ standard deviation. m= zα/2(σn) Solve the above equation for n.
A sample of n = 16 scores is selected from a population with μ = 80 with σ = 20. On average, how much error would be expected between the sample mean and the population mean?
Let x = age in years of a rural Quebec woman at the time of her first marriage. In the year 1941, the population variance of x was approximately σ2 = 5.1. Suppose a recent study of age at first marriage for a random sample of 51 women in rural Quebec gave a sample variance s2 = 3.4. Use a 5% level of significance to test the claim that the current variance is less than 5.1. (a) What is the level of significance? State the null and alternate hypotheses. H0: σ2 < 5.1; H1: σ2 = 5.1H0: σ2 = 5.1; H1: σ2 < 5.1 H0: σ2 = 5.1; H1: σ2 > 5.1H0: σ2 = 5.1; H1: σ2 ≠ 5.1 (b) Find the value of the chi-square statistic for the sample. (Round your answer to two decimal places.) What are the degrees of freedom? What assumptions are you making about the original distribution? We assume a exponential population distribution.We assume a normal population distribution. We assume a uniform population distribution.We assume a binomial population distribution. (c) Find or estimate the P-value…

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