A police department reports that the probabilities that 0, 1, 2, and 3 burglaries will be reported in a given day are 0.50, 0.41, 0.05, and 0.04, respectively. Find the standard deviation for the probability distribution. Round answer to the nearest hundredth.
see Mathguru's response to darius, who posted the exact same question.
To find the standard deviation for the probability distribution, you first need to calculate the expected value (mean) for the distribution.
The expected value is calculated by multiplying each possible outcome by its corresponding probability and summing them up. In this case, the possible outcomes are 0, 1, 2, and 3 burglaries reported, and their corresponding probabilities are 0.50, 0.41, 0.05, and 0.04, respectively.
Expected value = (0 * 0.50) + (1 * 0.41) + (2 * 0.05) + (3 * 0.04) = 0.41
Next, calculate the variance, which is the average of the squared differences between each outcome and the expected value, weighted by their probabilities.
Variance = [(0 - 0.41)^2 * 0.50] + [(1 - 0.41)^2 * 0.41] + [(2 - 0.41)^2 * 0.05] + [(3 - 0.41)^2 * 0.04] = 0.4841
Finally, calculate the standard deviation by taking the square root of the variance:
Standard deviation = √(0.4841) ≈ 0.6967 (rounded to the nearest hundredth)
Therefore, the standard deviation for the probability distribution is approximately 0.70.