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.
0.57
To find the standard deviation for the probability distribution, we first need to calculate the expected value (mean) of the distribution. Then we can use the formula for the standard deviation:
Standard Deviation = sqrt(∑(x - E(x))^2 * P(x))
where x is the random variable, E(x) is the expected value, and P(x) is the probability of that particular value.
1) Calculate the expected value (mean):
E(x) = (0 * 0.50) + (1 * 0.41) + (2 * 0.05) + (3 * 0.04)
E(x) = 0.41 + 0.10 + 0.12
E(x) = 0.63
2) Calculate the variance:
Variance = ∑(x - E(x))^2 * P(x)
Variance = (0 - 0.63)^2 * 0.50 + (1 - 0.63)^2 * 0.41 + (2 - 0.63)^2 * 0.05 + (3 - 0.63)^2 * 0.04
Variance = (0.63^2 * 0.50) + (0.37^2 * 0.41) + (1.37^2 * 0.05) + (2.37^2 * 0.04)
Variance = 0.19845 + 0.06013 + 0.09485 + 0.09106
Variance = 0.44449
3) Calculate the square root of the variance to find the standard deviation:
Standard Deviation = sqrt(0.44449)
Standard Deviation ≈ 0.67 (rounded to the nearest hundredth)
Therefore, the standard deviation for the probability distribution is approximately 0.67.
To find the standard deviation for the probability distribution, we need to calculate the variance first.
The variance (σ^2) of a probability distribution can be calculated using the formula:
σ^2 = ∑ (x - μ)^2 * P(x)
where:
- x represents the number of burglaries reported (0, 1, 2, 3),
- μ represents the mean of the probability distribution, and
- P(x) represents the probability of x burglaries being reported.
First, let's calculate the mean (μ) of the probability distribution:
μ = (0 * 0.50) + (1 * 0.41) + (2 * 0.05) + (3 * 0.04)
= 0 + 0.41 + 0.10 + 0.12
= 0.63
Now, let's calculate the variance (σ^2):
σ^2 = (0 - 0.63)^2 * 0.50 + (1 - 0.63)^2 * 0.41 + (2 - 0.63)^2 * 0.05 + (3 - 0.63)^2 * 0.04
= 0.3969 * 0.50 + 0.2209 * 0.41 + 2.1769 * 0.05 + 2.8561 * 0.04
= 0.19845 + 0.09067 + 0.108845 + 0.114244
= 0.512205
Finally, we can calculate the standard deviation (σ) by taking the square root (√) of the variance:
σ = √(0.512205)
≈ 0.72
Rounding the answer to the nearest hundredth, the standard deviation for the probability distribution is approximately 0.72.
Your data:
X P(X)
0 0.50
1 0.41
2 0.05
3 0.04
Here's a formula for the mean:
SUM[X * P(X)]
(Take each X times its respective P(X), then sum for a total. This will be your mean.)
Here's a formula for the standard deviation:
SQRT of {SUM[X^2 * P(X)] - mean^2}
(Square each X. Take each X^2 times its respective P(X), then sum for a total. Subtract mean^2 from this value. Then take the square root of that result. This will be your standard deviation.)
I hope this will help get you started.