Is r0 yes yes no no i1 n i n x the sample mean proved

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Is r=0? Yes Yes No No i=1 n i n X
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the sample mean. Proved that X, sample mean is an unbiased estimator of population mean θ . And also find its mean square error. (10 marks) Solution [ ] [ ] 1 1 i i n i n i X E X E n E X n n n θ θ = = = = = = 10. The quantity S 2 = 2 1 ( ) 1 i i n X X n = is called the sample variance and show that sample variance S is an unbiased estimator of population variance σ . (10 marks) [ ] [ ] [ ] [ ] 2 2 i n 1 i 2 2 i 2 n 1 i 2 i 2 n 1 i 2 i 2 X E - X E n X n X E S E 1) - (n 1) - (n ) X (X E S E 1 - n ) X (X S = = = = = = = We have,
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[ ] [ ] ( ) [ ] [ ] ( ) [ ] [ ] ( ) [ ] [ ] ( ) [ ] ( ) [ ] 2 2 2 2 2 2 2 2 2 2 2 2 2 2 i i 2 i 2 2 2 2 σ S E θ n σ n - θ σ n S E 1) - (n θ n σ X E ) X Var( X E and θ σ X E ) Var(X X E Y E Var(Y) Y E Y E - Y E Var(Y) = + + = + = + = + = + = + = = Therefore, Sample variance, S 2 is an unbiased estimator of population variance, 2 σ . 11. For any set at number x , ….., x prove algebraically that ( x – x ) = x - n x where x = x /n . (10 marks) ( ) x n - x x n x 2n - x x n x n x 2 - x x x x 2 - x ) x x x 2 (x ) x (x n 1 i 2 2 i 2 n 1 i 2 2 i 2 n 1 i 2 i n 1 i n 1 i 2 i n 1 i 2 i n 1 i 2 i 2 i n 1 i 2 i = = = = = = = = = + = + = + = + = 12. Write a method for determining when to stop generating new data to estimate a population mean. (10 marks) solution 1. Choose an acceptable value d for the standard deviation of the estimator. 2. Generate at least 100 data values. 3. Continue to generate additional data values, stopping when k data values is generated so that d k S , where S is the sample standard deviation based on those k values. 4. The estimate of θ is given by = = k 1 i i k X X . 1 n n 2 n i= 1 2 2 i= 1 i= 1 n i i i
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13. Write a procedure for determining when to stop generating new values to estimate a probability. (The data values are Berroulli random variables). (10 marks) Solution Bernoulli random variables X are generated, such that [ ] p) - (1 p ) Var(X and p X E p - 1 y probabilit with 0 p y probabilit with 1 X i i i = = = If n data values n 2 i, X .., , X X are generated, the estimate of p is = = n 1 i i n n X X So, Natural estimate of ) Var(X i is ( ) n n X - 1 X . 1. Choose an acceptable value d for the standard deviation of the estimator. 2. Generate at least 100 data values. 3. Continue to generate additional data values, stopping when k data values is generated so that ( ) [ ] d k X - 1 X k k , where S is the sample standard deviation based on those k values. 4. The estimate of p is k X , the average of the k data values. 14. If the first three data values are X1=5, X2=14, X3=9, and then find their sample mean and simple variance. (10 marks) Solution X 1 = 5, X 2 = 14, X 3 = 9 ( ) ( ) ( ) ( ) ( ) 3 61 2 19 - 3 28 3 4 81 S 2 81 5 - 2 19 2 S 3 28 2 19 - 9 3 1 2 19 X 2 19 2 9 5 X 5 X X - X 1 j S j 1 - 1 S 1 j X - X X X 2 2 3 2 2 2 3 2 1 2 ji 1 j 2 j 2 1 j j 1 j 1 j 1 j = + = = = = + = = + = = + + = + + = + + + + + 15. Suppose we are interested in estimating θ (F) =E[X] by using the sample mean