Final answer:
The expected value of the random variable is 2.1 and the standard deviation is 1.14. The expected value represents the average number of cars per household, while the standard deviation represents the variability or spread of the data around the expected value.
Step-by-step explanation:
To calculate the expected value of the random variable representing the number of cars per household, we multiply each value by its corresponding relative frequency and sum them up. In this case, the expected value is calculated as: (0 * 0.10) + (1 * 0.30) + (2 * 0.40) + (3 * 0.12) + (4 * 0.06) + (5 * 0.02) = 2.1.
To calculate the standard deviation, we need to find the variance first. The variance is calculated by summing up the squared differences between each value and the expected value, weighted by their relative frequencies. Then, the standard deviation is the square root of the variance. In this case, the variance is calculated as: ((0 - 2.1)^2 * 0.10) + ((1 - 2.1)^2 * 0.30) + ((2 - 2.1)^2 * 0.40) + ((3 - 2.1)^2 * 0.12) + ((4 - 2.1)^2 * 0.06) + ((5 - 2.1)^2 * 0.02) ≈ 1.29, and the standard deviation is √1.29 ≈ 1.14.
The expected value of the random variable represents the average number of cars per household, which in this case is 2.1. The standard deviation represents the variability or spread of the data around the expected value, which in this case is approximately 1.14. This means that most households have 2 cars, on average, and the majority of households fall within 1.14 cars of the average.