Role-playing games like Dungeons & Dragons use many different types of dice. Sup

Question

Role-playing games like Dungeons & Dragons use many different types of dice. Suppose that a eight-sided die has faces marked 1, 2, 3, 4, 5, 6, 7, 8. The intelligence of a character is determined by rolling this die twice and adding 1 to the sum of the spots. The faces are equally likely and the two rolls are independent.

What is the average (mean) intelligence for such characters?

How spread out are their intelligences, as measured by the standard deviation of the distribution? (Round your answer to four decimal places.)

Explanation / Answer

Here the mean value of the intelligence is computed as:

= 1 + (1/8)*(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8)

= 5.5

Therefore the mean intelligence score is 5.5

b) Now the standard deviation of the spread of intelligence scores is computed as:

First the second moment is computed as:

E(X2) = (1/8)*(12 + 22 + 32 + 42 + 52 + 62 + 72 + 82) = 25.5

Therefore, the variance is computed as: V(X) = E(X2) - (E(X))2 = 25.5 - 4.52 = 5.25

Now the standard deviation is computed as the square root of 5.25 which is 2.2913

Therefore the required standard deviation of the intelligence scores would be 2.2913

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