STAT 400 Lecture AL1 1 A balanced fair coin is tossed twice Let X denote the number of H s Construct the probability distribution of X Outcomes x TT 0 HT TH 1 HH 2 S HH HT TH TT X Spring 2015 Dalpiaz Answers for 2 1 2 2 2 3 2 1 1 Just for fun Suppose P H 0 60 P T 0 40 f x 1 4 1 2 1 4 1 00 0 Outcomes x f x TT 0 0 40 0 40 0 16 HT TH 1 0 60 0 40 0 40 0 60 0 48 HH 2 0 60 0 60 0 36 1 00 1 Suppose Homer Simpson has five coins 2 nickels 2 dimes and 1 quarter Let X denote the amount Bart gets if he steals two coins at random a Construct the probability distribution of X Outcomes x NN 0 10 ND DN DD 0 15 0 20 NQ QN 0 30 DQ QD 0 35 f x 5 4 20 2 2 2 2 8 5 4 5 4 20 2 1 2 5 4 20 2 1 1 2 4 5 4 5 4 20 2 1 2 5 4 5 4 20 2 1 1 2 4 0 1 0 4 0 1 0 2 0 2 1 0 OR N1 N2 D1 D2 Q x N1 0 10 0 15 0 15 0 30 0 10 N2 0 10 0 15 0 15 0 30 0 15 D1 0 15 0 15 0 20 0 35 0 20 D2 0 15 0 15 0 20 0 35 0 30 Q 0 30 0 30 0 35 0 35 0 35 f x 2 20 8 20 2 20 4 20 4 20 0 1 0 4 0 1 0 2 0 2 do not steal the same coin twice 1 0 OR Outcomes x NN 0 10 ND 0 15 DD 0 20 NQ 0 30 DQ 0 35 f x 2C 2 2 C 0 1C 0 5C 2 2 C1 2 C 1 1C 0 5C 2 2C 0 2 C 2 1C 0 5C 2 2 C1 2 C 0 1C 1 5C 2 2C 0 2 C 1 1C 1 5C 2 0 1 0 4 0 1 0 2 0 2 1 0 2 2 x f x x f x x X f x x f x 0 10 0 15 0 20 0 30 0 35 0 1 0 4 0 1 0 2 0 2 0 01 0 06 0 02 0 06 0 07 0 00144 0 00196 0 00004 0 00128 0 00338 0 0010 0 0090 0 0040 0 0180 0 0245 1 0 0 22 0 00810 0 0565 b Find the expected value of the amount that Bart gets E X X E X x f x 0 22 all x c Find the standard deviation SD X 2X Var X x X 2 f x 0 0081 all x OR 2X Var X x 2 f x 2 2X 0 0565 0 22 0 0565 0 0484 0 0081 all x X SD X 2 a 0 0081 0 09 Suppose a random variable X has the following probability distribution x f x 10 0 20 11 0 40 12 0 30 13 0 10 Find the expected value of X E X x f x x f x 10 0 2 2 0 11 0 4 4 4 12 0 3 3 6 13 0 1 1 3 1 0 11 3 X E X x f x 11 3 all x b Find the variance of X Var X x f x x X x X 2 f x 10 0 2 1 3 1 69 0 2 0 338 11 0 4 0 3 0 09 0 4 0 036 12 0 3 0 7 0 49 0 3 0 147 13 0 1 1 7 2 89 0 1 0 289 0 810 2X Var X x X 2 f x 0 81 all x OR x f x x2 f x 10 0 2 20 0 11 0 4 48 4 12 0 3 43 2 13 0 1 16 9 128 5 2X Var X x 2 f x E X 2 128 5 11 3 2 0 81 all x c Find the standard deviation of X SD X X SD X d 2X 0 9 Find the cumulative distribution function of X F x P X x x f x F x 10 11 12 13 0 2 0 4 0 3 0 1 0 2 0 6 0 9 1 0 F x 0 0 2 0 6 0 9 1 x 10 10 x 11 11 x 12 12 x 13 x 13 3 Suppose E X 7 SD X 3 a Y 2 X 3 Find E Y and SD Y SD Y 2 SD X 6 E Y 2 E X 3 17 W 5 2 X Find E W and SD W b E W 5 2 E X 9 3 SD Y 2 SD X 6 Suppose a discrete random variable X has the following probability distribution 1 x f x x 1 2 3 2 a Verify that this is a valid probability distribution f x 0 x f x 1 all x x first term 1 2 1 base x 1 1 2 1 1 2 1 b Find P X is divisible by 3 P X is divisible by 3 P 3 P 6 P 9 P 12 c 1 23 1 26 1 29 1 2 12 1 8 k 1 k 1 1 1 8 1 1 7 8 Find P X is divisible by 3 X is divisible by 2 P X is divisible by 3 X is divisible by 2 P X is divisible by 3 X is divisible by 2 P X is divisible by 2 P X is divisible by 6 P X is divisible by 2 P X is divisible by 2 P 2 P 4 P 6 P 8 1 2 4 6 8 2 2 2 2 k 1 4 1 1 1 1 k 1 1 4 1 1 4 1 3 P X is divisible by 6 P 6 P 12 P 18 P 24 1 6 12 18 24 2 2 2 2 k 1 64 1 1 1 1 k 1 1 64 1 1 P X is divisible by 3 X is divisible by 2 P X is divisible by 3 X is divisible by 2 P X is divisible by 2 P X is divisible by 6 P X is divisible by 2 1 63 1 1 21 3 64 1 63 d Find E X E X all x 1 x f x x x 1 2 x 1 E X 2 e 1 1 2 1 2 1 1 1 E X E X E X 2 2 E X 2 1 2 1 1 21 1 2 1 22 3 1 2 2 1 22 …
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