Let X be a random variable with mean and variance \u03c3 Find the expected value and

Let x be a random variable with mean and variance σ

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Let X be a random variable with mean 𝜇 and variance σ . Find the expected value and variances of random variables 3X 2X+5
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Example: Sample Mean & Variance Are Unbiased-1 X is a random variable with mean μ and variance σ 2 . Let X 1 , X 2 ,…, X n be a random sample of size n . Show that the sample mean ( X -bar) is an unbiased estimator of μ .
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Show that the sample variance ( S 2 ) is an unbiased estimator of σ 2 .
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Minimum Variance Unbiased Estimators If we consider all unbiased estimators of θ , the one with the smallest variance is called the minimum variance unbiased estimator (MVUE). If X 1 , X 2 ,…, X n is a random sample of size n from a normal distribution with mean μ and variance σ 2 , then the sample X - bar is the MVUE for μ . WHY?
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Standard Error of an Estimator
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Mean Squared Error Conclusion: The mean squared error (MSE) of the estimator is equal to the variance of the estimator plus the bias squared.
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Relative Efficiency The MSE is an important criterion for comparing two estimators. If the relative efficiency is less than 1, we conclude that the 1 st estimator is superior than the 2 nd estimator.
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Optimal Estimator A biased estimator can be preferred than an unbiased estimator if it has a smaller MSE. Biased estimators are occasionally used in linear regression. An estimator whose MSE is smaller than that of any other estimator is called an optimal estimator. Figure 7-8 A biased estimator that has a smaller variance than the unbiased estimator .
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Example-5 Suppose that the random variable X has the continuous uniform distribution Suppose that a random sample of n = 12 observations is selected from this distribution. What is the approximate probability distribution of X − 6 ? Find the mean and variance of this quantity.
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Exmple-5
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Example-6 A computer software package calculated some numerical summaries of a sample of data. The results are displayed here: (a) Fill in the missing quantities. (b) What is the estimate of the mean of the population from which this sample was drawn?
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Example-7 Let X 1 and X 2 be independent random variables with mean μ and variance σ2. Suppose that we have two estimators of μ: (a) Are both estimators unbiased estimators of μ? (b) What is the variance of each estimator?
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Example-7
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