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 variance for the number of shoplifting incidents during a given day.
To determine the variance for the number of shoplifting incidents during a given day, we first need to calculate the mean. The mean is calculated by multiplying each number of incidents by its corresponding probability and summing up the results:
Mean = (0*0.1) + (1*0.2) + (2*0.25) + (3*0.3) + (4*0.15)
Mean = 0 + 0.2 + 0.5 + 0.9 + 0.6
Mean = 2.2
Next, we calculate the variance using the formula:
Variance = Σ [(Xi - Mean)^2 * Probability(Xi)]
Variance = (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
Variance = 2.2
Variance = (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
Variance = 0 * 0.1 + 1.2 * 0.2 + 0.04 * 0.25 + 0.64 * 0.3 + 1.64 * 0.15
Variance = 0 + 0.24 + 0.01 + 0.192 + 0.246
Variance = 0.686
Therefore, the variance for the number of shoplifting incidents during a given day is 0.686.