First slide

Introduction to statistics

Question

If a variable takes values $0, 1, 2, \dots, n$ with frequencies $q^{n}, \frac{n}{1} q^{n - 1} p, \frac{n (n - 1)}{1 \cdot 2} q^{n - 2} p^{2}, \dots p^{n}$ , where $p + q = 1$ then the mean is

Moderate

A

np
B

nq
C

n(p+q)
D

None of these

Solution

Required mean,
$\Rightarrow \bar{x} = \frac{0 \cdot q^{n} + 1 \cdot \frac{n}{1} q^{n - 1} p + 2 \cdot \frac{n (n - 1)}{2!} q^{n - 2} p^{2} + \dots + n \cdot p^{n}}{q^{n} + \frac{n}{1} q^{n - 1} p + \frac{n (n - 1)}{2!} q^{n - 2} p^{2} + \dots + p^{n}}$
$\begin{array}{l} \Rightarrow \bar{x} = \frac{0 \cdot^{n} C_{0} q^{n} p^{0} + 1 \cdot^{n} C_{1} q^{n - 1} p + \dots + n \cdot^{n} C_{n} q^{0} p^{n}}{^{n} C_{0} q^{n} p^{0} +^{n} C_{1} q^{n - 1} p^{1} + \dots +^{n} C_{n} q^{n - n} p^{n}} \\ \Rightarrow \bar{x} = \frac{\sum_{r = 0}^{n} r \cdot^{n} C_{r} q^{n - r} p^{r}}{\sum_{r = 0}^{n}^{n} C_{r} q^{n - r} p^{r}} = \frac{\sum_{r = 0}^{n} r \cdot {\frac{n}{r}}^{n - 1} C_{r - 1} q^{n - r} p \cdot p^{r - 1}}{\sum_{r = 0}^{n}^{n} C_{r} q^{n - r} p^{r}} \end{array}$
$\begin{array}{l} \Rightarrow \bar{x} = \frac{np [\sum_{r = 1}^{n} n - 1}{\sum_{r - 1} p^{r - 1} q^{(n - 1) - (r - 1)}]} \\ \Rightarrow \bar{x} = \frac{np (q + p)^{n - 1}}{(q + p)^{n}} [^{n - r} p^{r} \\ ∴ \bar{x} = np \end{array}$
$(∵ q + p = 1)$

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