Question

If a variable X takes values 0, 1, 2,..., n with frequencies ⁿC₀, ⁿC₁, ⁿC₂ , ... , ⁿC_n, then write variance X.

Answer 1

\[x = \frac{\sum^n_{i = 0} x_i f_i}{\sum^n_{i = 0} f_i} = \frac{0 \times ^{n}{}{C}_o + 1 \times^{n}{}{C}_1 + . . . + n \times ^{n}{}{C}_n}{T^{n}{}{C}_o +^{n}{}{C}_1 + . . . +^{n}{}{C}_n}\]

\[ \Rightarrow x = \frac{n \times 2^{n - 1}}{\frac{2^n}{n + 1}}\]

\[ = \frac{n\left( n + 1 \right)}{2}\]

\[ \therefore Var(X) = \sigma^2 \]

\[ = \frac{1}{n} \sum^n_{i = 0} \left( x_i - x \right)^2 \]

\[ = \frac{1}{n} \left[ \left( 0 + 1 + 2 + . . . . + n \right) - nx \right]^2 \]

\[ \Rightarrow \sigma^2 = \frac{1}{n} \left[ \frac{n\left( n + 1 \right)}{2} - \frac{n \times n\left( n + 1 \right)}{2} \right]^2 \]

\[ = \frac{1}{n} \left[ \frac{n\left( n + 1 \right)}{2}\left( 1 - n \right) \right]^2 \]

\[ = \frac{n^2}{4n} \left( n + 1 \right)^2 \left( n - 1 \right)^2 \]