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Lab4WithSol - Statistics 103 Lab 4 LAB PROBLEMS 1 Book...

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Statistics 103: Lab 4 LAB PROBLEMS 1. Book problem 8.3 (This one is a bit tricky. You can assume the two students are chosen with replace- ment. Hint: use the Binomial PMF to figure out the marginals; figure out which entries of the joint PMF are zero; then fill in the rest). 2. Book problem 8.10. 3. Book problem 8.14. 4. Suppose X and Y are independent random vari- ables with E ( X ) = μ 1 , var ( X ) = σ 2 1 , E ( Y ) = μ 2 , and var ( Y ) = σ 2 2 . Compute E (2 X - 0 . 1 Y + 5), E ( μ 1 X + σ 2 2 Y + μ 2 ), var ( - 3 Y + μ 2 X + 999), SD (5 X - 14 Y ). answer: 2 μ 1 - 0 . 1 μ 2 +5 ; μ 2 1 + σ 2 2 μ 2 + μ 2 ; 9 σ 2 2 + μ 2 2 σ 2 1 ; p 25 σ 2 1 + 196 σ 2 2 PRACTICE PROBLEMS 1. Let us assume that X 1 , X 2 , X 3 , . . . , X n are indepen- dent normal random variables mean μ and variance σ 2 . That is, X i N ( μ, σ 2 ) for i = 1 , 2 , 3 , . . . , n . Let Y = n i =1 X i . Find EY , E ( ¯ X ), E ( Y - n ¯ X ), var ( Y ), var ( ¯ X ), var ( Y - n ¯ X ), var ( μ + σ 2 ), Distribution of Y , Dis- tribution of ¯ X . For var ( Y - n ¯ X ), simplify Y - n ¯ X first before solving the question. answer: ; μ ; 0 ; 2 ; σ 2 n , 0 , 0 , N ( nμ, nσ 2 ) , N ( μ, σ 2 n ) 2. Suppose that you have a box which contains cards. Each card has either “0” or “1” is written on it. Suppose further that when drawing a card at ran- dom, the probability that you see “1” on it is p . Now randomly draw n cards with replacement and let X 1 , X 2 , . . . , X n denote the n card numbers. Then, we say that X i has a Binomial distribution based on 1 trial with probability of success p , writ- ten X i Bin (1 , p ). Now, consider Y = n i =1 X i . Then, Y is the total of the numbers written on n cards. It is known that Y has a Binomial distribu- tion with parameters ( n, p ), or Y Bin ( n, p ).
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