If a variable takes values 0, 1, 2, ..., n wi | vedclass

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Question

(B) The total frequency $N = \sum_{r=0}^n {^nC_r} = 2^n$. The sum of the products of values and frequencies is $\sum_{r=0}^n r \cdot {^nC_r} = 0 \cdot

Vedclass · Accepted Answer

$\frac{n}{2}$

Vedclass · Answer

(B) The total frequency $N = \sum_{r=0}^n {^nC_r} = 2^n$. The sum of the products of values and frequencies is $\sum_{r=0}^n r \cdot {^nC_r} = 0 \cdot {^nC_0} + 1 \cdot {^nC_1} + 2 \cdot {^nC_2} + ... + n \cdot {^nC_n}$. Using the identity $r \cdot {^nC_r} = n \cdot {^{n-1}C_{r-1}}$,the sum becomes $\sum_{r=1}^n n \cdot {^{n-1}C_{r-1}} = n \sum_{k=0}^{n-1} {^{n-1}C_k} = n \cdot 2^{n-1}$. Thus,the mean $\bar{x} = \frac{\sum f_i x_i}{N} = \frac{n \cdot 2^{n-1}}{2^n} = \frac{n}{2}$.

$X = x_i$	$0$	$1$	$2$	$3$
$P(X = x_i)$	$k$	$3k$	$3k$	$k$

$X=x$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$
$P(X=x)$	$0$	$K$	$2K$	$2K$	$3K$	$K^2$	$2K^2$	$7K^2+K$

If a variable takes values $0, 1, 2, ..., n$ with frequencies proportional to the binomial coefficients $^nC_0, ^nC_1, ..., ^nC_n$,find the mean of the distribution.

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