The variance of the first 50 even natural num | vedclass

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(D) The first $50$ even natural numbers are $2, 4, 6, \ldots, 100$.These can be written as $2(1), 2(2), 2(3), \ldots, 2(50)$.The variance of a set of

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$833$

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(D) The first $50$ even natural numbers are $2, 4, 6, \ldots, 100$.These can be written as $2(1), 2(2), 2(3), \ldots, 2(50)$.The variance of a set of numbers $x_i$ multiplied by a constant $k$ is given by $	ext{Var}(kx_i) = k^2 	ext{Var}(x_i)$.Here,$k = 2$ and the set is the first $50$ natural numbers $(1, 2, 3, \ldots, 50)$.The variance of the first $n$ natural numbers is $\frac{n^2 - 1}{12}$.For $n = 50$,$	ext{Var} = \frac{50^2 - 1}{12} = \frac{2500 - 1}{12} = \frac{2499}{12}$.Therefore,the variance of the first $50$ even natural numbers is $2^2 	imes \frac{2499}{12} = 4 	imes \frac{2499}{12} = \frac{2499}{3} = 833$.

The variance of the first $50$ even natural numbers is

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