Step 1: Identify the first 50 even natural numbers. The first 50 even natural numbers are: \[ 2, 4, 6, \dots, 100.\] This sequence forms an arithmetic progression with:
- First term \( a = 2 \),
- Common difference \( d = 2 \),
- Number of terms \( n = 50 \).
Step 2: Calculate the sum and mean. The sum of this arithmetic progression is calculated as: \[S = \frac{n}{2} \left( 2a + (n-1)d \right) = \frac{50}{2} \left( 2(2) + (50-1)2 \right) = 25(4 + 98) = 2550.\] The mean is determined by: \[\text{Mean} = \frac{S}{n} = \frac{2550}{50} = 51.\]Step 3: Compute the sum of squares. The sum of the squares of these numbers is found using the formula: \[\sum_{k=1}^{n} (2k)^2 = 4 \sum_{k=1}^{n} k^2 = 4 \times \frac{n(n+1)(2n+1)}{6}.\] With \( n = 50 \), the sum of squares is: \[4 \times \frac{50(50+1)(2 \times 50+1)}{6} = 4 \times \frac{50 \times 51 \times 101}{6} = 171700.\]Step 4: Determine the variance. We calculate \( E(X^2) \) and the variance: \[E(X^2) = \frac{171700}{50} = 3434,\] \[\text{Variance} = E(X^2) - (\text{Mean})^2 = 3434 - 51^2 = 3434 - 2601 = 833.\] Consequently, the variance is \( 833 \).