Find the standard deviation for the given probability distribution. Round to the nearest hundredth.
x P(x) 0 0.26 1 0.27 2 0.18 3 0.12 4 0.17
To find the standard deviation for a given probability distribution, we need to follow these steps:
1. Calculate the expected value (mean) of the distribution.
2. Calculate the variance of the distribution.
3. Take the square root of the variance to find the standard deviation.
Step 1: Calculate the expected value (mean)
To find the expected value, we need to multiply each value of x by its corresponding probability and then sum them up.
Expected value (mean) = (0 * 0.26) + (1 * 0.27) + (2 * 0.18) + (3 * 0.12) + (4 * 0.17) = 0 + 0.27 + 0.36 + 0.36 + 0.68 = 1.67
Step 2: Calculate the variance
The variance is the average of the squared differences between each value and the mean. We need to calculate the squared differences, multiply them by their corresponding probabilities, and then sum them up.
Variance = [(0 - 1.67)^2 * 0.26] + [(1 - 1.67)^2 * 0.27] + [(2 - 1.67)^2 * 0.18] + [(3 - 1.67)^2 * 0.12] + [(4 - 1.67)^2 * 0.17]
= [(-1.67)^2 * 0.26] + [(-0.67)^2 * 0.27] + [(0.33)^2 * 0.18] + [(1.33)^2 * 0.12] + [(2.33)^2 * 0.17]
= 1.1066 + 0.1199 + 0.0162 + 0.2117 + 0.9613
= 2.4157
Step 3: Calculate the standard deviation
Now, we can take the square root of the variance to get the standard deviation.
Standard deviation = √(2.4157) ≈ 1.55 (rounded to the nearest hundredth)
Therefore, the standard deviation for the given probability distribution is approximately 1.55.