I pick a random integer from 0 to 99 (inclusive) and add the digits to- gether. What is my expected sum?
Answers
Given : random integer from 0 to 99
add the digits to- gether.
To Find : expected sum
Solution:
0 to 99 - Total 100 numbers
Sum
Sum 0 = 0
Sum 1 = 1 , 10
Sum 2 = 2 , 11 , 20
Sum 3 = 3 , 12 , 21 , 30
Sum 4 = 4 , 13 , 22 , 31 , 40
Sum 5 = 5 , 14 , 23 , 32 , 41 , 50
Sum 6 = 6 , 15 , 24 , 33 , 42 , 51 , 60
Sum 7 = 7 , 16 , 25 , 34 , 43 , 52 , 61 , 70
Sum 8 = 8 , 17 , 26 , 35 , 44 , 53 , 62 , 71 , 80
Sum 9 = 9 , 18 , 27 , 36 , 45 , 54 , 63 , 72 , 81 , 90
Sum 10 = 19 , 28 , 37 , 46 , 55 , 64 , 73 , 82 , 91
Sum 11 = 29 , 38 , 47 , 56 , 65 , 74 , 83 , 92
Sum 12 = 39 , 48 , 57 , 66 , 75 , 84 , 93
Sum 13 = 49 , 58 , 67 , 76 , 85 , 94
Sum 14 = 59 , 68 , 77 , 86 , 95
Sum 15 = 69 , 78 , 87 , 96
Sum 16 = 79 , 88 , 97
Sum 17 = 89 , 98
Sum 18 = 99
expected sum = E(X) .P(X)
= (1/100) ( 0 * 1 + 1 * 2 + 2 * 3 + 3 * 4 + 4 * 5 + 5 * 6 + 6 * 7 + 7 * 8 + 8 * 9 + 9 * 10 + 10 * 9 + 11 * 8 + 12 * 7 + 13 * 6 + 14 * 5 + 15 * 4 + 16 * 3 + 17 * 2 + 18 *1)
= (1/100) ( 0 + 2 + 6 + 12 + 20 + 30 + 42 + 56 + 72 + 90 + 90 + 88 + 84 + 78 + 70 + 60 + 48 + 34 + 18)
= 900/100
= 9
expected sum is 9
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