Assume that the table below represents the distribution of X. Note that X is a discrete random variable. The second column reports the number of observations for each value of X. Based on this information, what is the variance of X?
X n
20 10
25 10
30 5
15 10
10 5
To find the variance of a discrete random variable, you need to follow these steps:
1. Calculate the mean (μ) of the random variable X.
2. Subtract the mean from each value of X to obtain the deviations from the mean (X - μ).
3. Square each deviation (X - μ)^2.
4. Multiply each squared deviation by the corresponding frequency for that value of X.
5. Sum up all the squared, weighted deviations to get the sum of squared deviations (Σ(X - μ)^2).
6. Divide the sum of squared deviations by the total number of observations (n).
7. The result is the variance of X.
Let's perform these calculations based on the given table:
1. Calculate the mean (μ):
mean (μ) = (20 * 10 + 25 * 10 + 30 * 5 + 15 * 10 + 10 * 5) / (10 + 10 + 5 + 10 + 5) = 20
2. Subtract the mean from each value of X to obtain the deviations from the mean (X - μ):
X - μ = [20 - 20, 25 - 20, 30 - 20, 15 - 20, 10 - 20] = [0, 5, 10, -5, -10]
3. Square each deviation (X - μ)^2:
(X - μ)^2 = [0^2, 5^2, 10^2, (-5)^2, (-10)^2] = [0, 25, 100, 25, 100]
4. Multiply each squared deviation by the corresponding frequency for that value of X:
[0 * 10, 25 * 10, 100 * 5, 25 * 10, 100 * 5] = [0, 250, 500, 250, 500]
5. Sum up all the squared, weighted deviations to get the sum of squared deviations (Σ(X - μ)^2):
Σ(X - μ)^2 = 0 + 250 + 500 + 250 + 500 = 1500
6. Divide the sum of squared deviations by the total number of observations (n):
Variance = Σ(X - μ)^2 / n = 1500 / (10 + 10 + 5 + 10 + 5) = 1500 / 40 = 37.5
Therefore, the variance of X is 37.5.