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.
1.46
4.1
6.3
1.2083
2.0248
To determine the variance for the number of shoplifting incidents during a given day, we first need to calculate the mean (expected value) of the distribution.
Mean (μ) = Σ(x * P(x))
= 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
Now, we can calculate the variance using the formula:
Variance (σ^2) = Σ[(x - μ)^2 * P(x)]
= (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
Therefore, the variance for the number of shoplifting incidents during a given day is 1.46.