JAMIA 2022 — Mathematics PYQ

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Question

JAMIA 2022 — Mathematics PYQ

JAMIA | Mathematics | 2022

The variance of first 50 even natural numbers is

Choose the correct answer:

Explanation

Step 1: Identify the Sequence

The first 50 even natural numbers are:

2, 4, 6, 8, \dots, 100

This is an arithmetic progression where the number of terms ( $n$ ) is 50.

Step 2: Use the Variance Property

Let $X$ be the set of the first 50 natural numbers: $\{1, 2, 3, \dots, 50\}$ .

The set of the first 50 even natural numbers can be written as $2X$ : $\{2 \cdot 1, 2 \cdot 2, 2 \cdot 3, \dots, 2 \cdot 50\}$ .

A fundamental property of variance is:

\text{Var}(kX) = k^2 \cdot \text{Var}(X)

In this case, $k = 2$ , so:

\text{Var}(\text{Even Numbers}) = 2^2 \cdot \text{Var}(\text{First 50 Natural Numbers})

\text{Var}(\text{Even Numbers}) = 4 \cdot \text{Var}(1, 2, \dots, 50)

Step 3: Variance of First $n$ Natural Numbers

The formula for the variance of the first $n$ natural numbers is:

\text{Var}(n) = \frac{n^2 - 1}{12}

For $n = 50$ :

\text{Var}(1, 2, \dots, 50) = \frac{50^2 - 1}{12}

\text{Var}(1, 2, \dots, 50) = \frac{2500 - 1}{12} = \frac{2499}{12}

Step 4: Calculate the Final Variance

Now, multiply by $k^2 = 4$ :

\text{Variance} = 4 \times \frac{2499}{12}

\text{Variance} = \frac{2499}{3}

\text{Variance} = 833