Question

Compute the mean, standard deviation, and variance for the probability distribution below. Round each value to the nearest hundredth (two decimal places). X 1 4 6 18 P(X) 0.075 0.65 0.05 0.225 μ =7.03 0 = 5.99 G²=35.82 Ou= 7.03, o = 5.99, 0² = 35.82 Oμ = 9.03, 0 = 6.79, 0² = 45.82 μ5.03, 03.99, ²=30.82 Oμ = 6.03, o = 4.99, 0² = 33.82

Answer 1

The mean (μ) of the probability distribution is 5.99, the standard deviation (σ) is approximately 7.03, and the variance (σ²) is about 49.42.

Step-by-step explanation:

To compute the mean (μ) of the probability distribution, we multiply each value of X by its respective probability P(X) and sum these products. So,

μ = (1 * 0.075) + (4 * 0.65) + (6 * 0.05) + (18 * 0.225) = 0.075 + 2.6 + 0.3 + 4.05 = 7.03.

The variance (σ²) is calculated as the sum of the squared differences between each value of X and the mean, multiplied by their respective probabilities.

σ² = (1 - 7.03)² * 0.075 + (4 - 7.03)² * 0.65 + (6 - 7.03)² * 0.05 + (18 - 7.03)² * 0.225

= 35.82.

Finally, the standard deviation (σ) is the square root of the variance:

σ = √35.82 ≈ 5.99.

These calculations demonstrate the statistical measures for this probability distribution. The mean represents the average value, indicating the expected value of X. The variance measures the spread of the distribution from the mean, while the standard deviation gives a clearer understanding of how spread out the values are around the mean. In this case, the mean (5.99), standard deviation (7.03), and variance (49.42) describe the central tendency and variability within this probability distribution.

Answer 2

To find the mean, multiply each value by its corresponding probability and sum them up. To find the variance, subtract the mean from each value, square the result, multiply it by its probability, and sum them up. The standard deviation is the square root of the variance.

Step-by-step explanation:

The mean of a probability distribution is calculated by multiplying each value by its corresponding probability and then summing them up. In this case, the mean (μ) can be calculated as (1 × 0.075) + (4 × 0.65) + (6 × 0.05) + (18 × 0.225) = 7.03.

To calculate the variance of a probability distribution, we need to subtract the mean from each value, square the result, multiply it by its corresponding probability, and then sum them up. The variance (σ^2) can be calculated as [(1 - 7.03)^2 × 0.075] + [(4 - 7.03)^2 × 0.65] + [(6 - 7.03)^2 × 0.05] + [(18 - 7.03)^2 × 0.225] = 35.82.

The standard deviation (σ) can be found by taking the square root of the variance. Therefore, the standard deviation is √35.82 ≈ 5.99.