Problem 2 - Compute the expected value, the variance and thestandard deviation o

Question

Problem 2 - Compute the expected value, the variance and thestandard deviation of X.

a)    You have 50 coins that have "1" on one sideand "2" on the other, and 20 dice. You throw all of them high inthe air, they fall on the floor and you count the sum of thenumbers each coin and die has on top, the result is X.

b)    There are 10 balls, 3 red and 7 blue. You pickat random 5 balls. X is the number of blue balls.

c) There are 1000 boxes. Each box has a number on its lid and onits bottom, and these two numbers are identical, however, each boxhas a different number. We remove the lids and put them back on theboxes in random; X is the resulting num- ber of boxes with the samenumber on the lid and on the bottom.

Explanation / Answer

a) Expected value for just the coin is 1.5 with variance = 0.25 andstandard deviation of 0.5 Expected value for die is 3.5 with variance = 2.92 and standarddeviation of 1.7 Expected value for sum is 1+5+3.5 = 5.0, variance = 0.25+2.92 =3.2, standard deviation = 3.2 = 1.78 b) 70% are blue Expected Value = .7*5 = 3.5 variance = 5(0.7)(1-0.7) = 5*.7*.3 = 1.05 Standard Deviation = 1.05 = 1.025 c) Hardest one. Because of the large number of boxes, we canapproximate as binomial. Therefore Expected Value = np = 1000(1/1000) = 1 Variance = np(1-p) = 1000(1/1000)(999/1000) = 0.999 Standard deviation = 0.999 = .9995

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