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sta 2006 0809_addition_example_2

sta 2006 0809_addition_example_2 - STA2006 Basic Concepts...

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STA2006 Basic Concepts in Statistics II (2008-09, 2nd term) Additional Example 2 1. Let X 1 , X 2 , ..., X N be iid Binomial ( n, p ) where both n and p are unknown. Use method of moments to estimate p and n . 2. Let X i Gamma ( α, β ) , i = 1 , ..., n be independent random variables. Assuming α known, (a) Find the MLE estimator of β . Is the MLE unbiased? (b) Denote the MLE of β by ˆ β . Is 1 ˆ β an unbiased estimator of 1 β ? 3. Consider N independent random variables having identical binomial distributions with the parameters θ and n = 3. If n 0 of them take on the value 0, n 1 take on the value 1, n 2 take on the value 2, and n 3 take on the value 3, use the method of moments to find a formula for estimating θ . 4. A random sample of 121 observations ( n = 121) from a normal population produced the following data: 121 i =1 x i = 2734 121 i =1 ( x i - x ) 2 = 8421 (a) Calculate the sample mean and sample standard deviation. (b) Construct a 90% confidence interval for the population mean.
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