5 solution.pdf - STAT244 | HW5 solutions | Due Total 100...

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STAT244 | HW5 solutions | Due 2/16/2017 Total: 100 pts. Notation : 1. f ( t ) denotes the (probability) density (function) of random variable at t , F ( t ) - the (cumulative) distribution (function); 2. Φ denotes the distribution of N (0 , 1) ; 3. If n 2 N , i.e., n is an integer, [ n ] denotes 1 , 2 , . . . , n - 1 , n (sometimes denoted as 1 , n ); 4. l ( | x ) will denote the log-likelihood for parameter and data x ; 5. ˆ MLE stands for the MLE of ; 6. 1 A ( x ) denotes a function that is 1 whenever x 2 A , and 0 otherwise. Problem 1 [15 pts] Suppose X has a Beta ( a, b ) density. 1. [9 pts] Use the normal approximation to find all three of P | X - . 6 | < . 001 , P | X - . 6 | < . 01 and P | X - . 6 | < . 1 when X is (a) Beta (3 , 2) , (b) Beta (30 , 20) , (c) Beta (300 , 200) . Present the results in a 3 3 table. Solution . X Beta ( a, b ) ) μ , E [ X ] = a a + b ; σ 2 , Var [ X ] = ab ( a + b ) 2 ( a + b + 1) For large a and b , X is also approximately normal, Y N ( μ, σ 2 ) . Given standard normal Z N (0 , 1) with distribution Φ , for any x 2 R , P | Y - μ | < x = P σ | Z | < x = P | Z | < x/ σ = Φ ( x/ σ ) - Φ ( - x/ σ ) [3 pts] Using this result, we can construct a 3 3 table [6 pts] with the desired probabilities for P | X - . 6 | < x : x = . 001 x = . 01 x = . 1 a = 3 , b = 2 . 004 . 0399 . 3829 a = 30 , b = 20 . 0116 . 1159 . 8551 a = 300 , b = 200 . 0364 . 3523 1 2. [6 pts] For case (a) only, find the same three probabilities exactly directly from the beta density by integration. Solution . For a = 3 , b = 2 the density is f ( x ) = Γ (5) Γ (3) Γ (2) x 2 (1 - x ) = 12( x 2 - x 3 ) Let x - = . 6 - x and x + = . 6 + x , with x as in the previous item. We can compute the exact probabilities by computing the integral: P | X - . 6 | < x = 12 Z x + x - ( x 2 - x 3 ) dx = 4 x 3 + - 3 x 4 + - 4 x 3 - + 3 x 4 - [3 pts] We get results [3 pts] that closely resemble the ones in the previous table x = . 001 x = . 01 x = . 1 a = 3 , b = 2 . 0035 . 0346 . 3392 Problem 2 [13 pts] Suppose there are n voters and n is even. Suppose voters vote independently of one another, each with probability p of voting for candidate A and probability 1 - p of voting for candidate B . Let X be the total vote for A . 1. [2 pts] What is the probaiblity of a tie if p = . 5 and n = 20 ?
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