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(A) The given frequencies are the terms of the binomial expansion of $(q + p)^n$.The mean $\bar{x}$ is given by the formula $\bar{x} = \frac{\sum_{r=0

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$np$

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(A) The given frequencies are the terms of the binomial expansion of $(q + p)^n$.The mean $\bar{x}$ is given by the formula $\bar{x} = \frac{\sum_{r=0}^{n} r f_r}{\sum_{r=0}^{n} f_r}$.Here,$f_r = {^nC_r} q^{n-r} p^r$.The denominator is $\sum_{r=0}^{n} {^nC_r} q^{n-r} p^r = (q + p)^n = 1^n = 1$.The numerator is $\sum_{r=0}^{n} r {^nC_r} q^{n-r} p^r$.Since $r {^nC_r} = n {^{n-1}C_{r-1}}$,the numerator becomes $\sum_{r=1}^{n} n {^{n-1}C_{r-1}} q^{(n-1)-(r-1)} p^r$.$= np \sum_{r=1}^{n} {^{n-1}C_{r-1}} q^{(n-1)-(r-1)} p^{r-1}$.Let $k = r-1$,then the sum is $np \sum_{k=0}^{n-1} {^{n-1}C_k} q^{(n-1)-k} p^k = np(q + p)^{n-1} = np(1)^{n-1} = np$.Thus,the mean is $\frac{np}{1} = np$.

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

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