Suppose that for a given computer salesperson, the probability distribution of x = Number of systems sold in one week is given by the following table
x 1 2 3 4 5 6 7 8
p(x) 0.04 0.10 0.13 0.30 0.29 0.12 0.01 0.01
(a) Find the mean value of x (the mean number of systems sold).
(b) Find the variance and standard deviation of x. (Round your standard deviation to four decimal
places.)
(c) What is the probability that the number of systems sold is within 1 standard deviation of its mean
value?
(d) What is the probability that the number of systems sold is more than 2 standard deviations from the
mean?
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(a) The mean value of x can be calculated by multiplying each value of x by its corresponding probability and summing the results.
Mean = (1*0.04) + (2*0.10) + (3*0.13) + (4*0.30) + (5*0.29) + (6*0.12) + (7*0.01) + (8*0.01)
Mean = 0.04 + 0.20 + 0.39 + 1.20 + 1.45 + 0.72 + 0.07 + 0.08
Mean = 4.15
The mean number of systems sold is 4.15.
(b) The variance of x can be calculated by first finding the squared difference between each value of x and the mean, multiplying each squared difference by its corresponding probability, and summing the results. The standard deviation is simply the square root of the variance.
Variance = (1-4.15)^2 * 0.04 + (2-4.15)^2 * 0.10 + (3-4.15)^2 * 0.13 + (4-4.15)^2 * 0.30 + (5-4.15)^2 * 0.29 + (6-4.15)^2 * 0.12 + (7-4.15)^2 * 0.01 + (8-4.15)^2 * 0.01
Variance = 4.795
Standard deviation = sqrt(Variance)
Standard deviation ≈ 2.1909
The variance of x is 4.795 and the standard deviation is approximately 2.1909.
(c) To find the probability that the number of systems sold is within 1 standard deviation of its mean value, we need to calculate the probability of x being between the mean minus 1 standard deviation and the mean plus 1 standard deviation.
Probability = p(3) + p(4) + p(5) + p(6) ≈ 0.13 + 0.30 + 0.29 + 0.12
Probability ≈ 0.84
There is approximately an 84% probability that the number of systems sold is within 1 standard deviation of its mean value.
(d) To find the probability that the number of systems sold is more than 2 standard deviations from the mean, we need to add the probabilities of all values of x greater than the mean plus 2 standard deviations.
Probability = p(7) + p(8) ≈ 0.01 + 0.01
Probability ≈ 0.02
There is approximately a 2% probability that the number of systems sold is more than 2 standard deviations from the mean.
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