The formula for calculating the standard deviation of a probability distribution is as follows:
Standard deviation = √[Σ((xi - μ)^2) * P(xi)]
Where:
xi = each value in the distribution
μ = mean of the distribution
P(xi) = probability of each value
First, calculate the mean of the distribution:
Mean (μ) = Σ(xi * P(xi)) = (0*0.1) + (1*0.2) + (2*0.25) + (3*0.3) + (4*0.15) = 0 + 0.2 + 0.5 + 0.9 + 0.6 = 2.2
Next, calculate the standard deviation:
Standard deviation = √[(0-2.2)^2*0.1 + (1-2.2)^2*0.2 + (2-2.2)^2*0.25 + (3-2.2)^2*0.3 + (4-2.2)^2*0.15]
= √[(2.2)^2 * 0.1 + (1.2)^2 * 0.2 + (0.2)^2 * 0.25 + (0.8)^2 * 0.3 + (1.8)^2 *0.15]
= √[(4.84)*0.1 + (1.44)*0.2 + (0.04)*0.25 + (0.64)*0.3 + (3.24)*0.15]
= √[0.484 + 0.288 + 0.01 + 0.192 + 0.486]
= √1.46
≈ 1.21
Therefore, the standard deviation for the number of shoplifting incidents during a given day is approximately 1.21.
The following table lists the probability distribution of the number of shoplifting incidents that occur during a day at a certain shopping center:
Number of shoplifting incidents 0 1 2 3 4
Probability 0.1 0.2 0.25 0.3 0.15
Determine the standard deviation for the number of shoplifting incidents during a given day.
1.43
2.51
1.1
2.02
1.21
1 answer