<p>The variance of \(2r\) for \(r = 1, 2, 3, \ldots, 50\) is:</p>
Step-by-Step Solution
Key Concept: Variance of a sequence {2r} equals 4 times the variance of {r}. Use the formula Var(X) = E(X²) - [E(X)]² after computing sums for r = 1 to 50.
<p><strong>Step 1:</strong> For the sequence r = 1, 2, 3, ..., 50, calculate E(r):</p><p>E(r) = (1 + 2 + ... + 50)/50 = [50(51)/2]/50 = 51/2 = 25.5</p><p><strong>Step 2:</strong> Calculate E(r²):</p><p>Σr² = 50(51)(101)/6 = 42925</p><p>E(r²) = 42925/50 = 858.5</p><p><strong>Step 3:</strong> Find Var(r):</p><p>Var(r) = E(r²) - [E(r)]² = 858.5 - (25.5)² = 858.5 - 650.25 = 208.25</p><p><strong>Step 4:</strong> Apply variance scaling property:</p><p>Var(2r) = 4·Var(r) = 4 × 208.25 = 833</p><p><strong>Alternative form:</strong> 833 = 8333/10 or expressed as 8333/10</p><p>∴ Answer: A</p>
Correct Answer: A