After an analysis of incoming emails, the manager of an accounting firm determined the probability distribution of the number of pages per email as follows: [table]
x | 1 | 2 | 3 | 4
p(x) | 0.30 | 0.40 | 0.25 | 0.05
[/table] Using the probability distribution above, calculate the standard deviation of the number of pages per email. (2 DP)

Question

BlackTom AI · Accepted Answer

The question provides a discrete probability distribution for the number of pages per email and asks for the standard deviation of the number of pages.
First, restate the distribution: X takes values 1, 2, 3, 4 with probabilities 0.30, 0.40, 0.25, and 0.05 respectively.
Next, compute the mean (expected value) μ = E[X] = 1(0.30) + 2(0.40) + 3(0.25) + 4(0.05) = 0.30 + 0.80 + 0.75 + 0.20 = 2.05.
Then, compute E[X^2] = 1^2(0.30) + 2^2(0.40) + 3^2(0.25) + 4^2(0.05) = 1(0.30) + 4(0.40) + 9(0.25) + 16(0.05) = 0.30 + 1.60 + 2.25 + 0.80 = 4.95.
The variance is Var(X) = E[X^2] − (E[X])^2 = 4.95 − (2.05)^2 = 4.95 − 4.2025 = 0.7475.
Finally, the standard deviation is the square root of the variance: SD(X) = sqrt(0.7475) ≈ 0.8647, which is approximately 0.86 when rounded to two decimal places.
Note that this result matches the common rounding convention for the given numeric answer format.

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