Suppose a discrete random variable X takes on the values $0, 1, 2, ,n$ with frequencies proportional to binomial coefficients $\binom{n}{0}, \binom{n}{1},....,\binom{n}{n}$ Then the mean (m ) and the variance ($s_{2}$ ) of the distribution are?
I differentiated the binominal expansion and took $x=1$ which concluded to $\sum x_{i}= n2^{n-1}$ .Therefore the mean should be $ \frac{\sum x_{i}}{n}=2^{n-1} $. But, the mean is $\frac{n}{2}$.
Can anyone help me, thanks in advance.
The total frequency isn't $n$. In fact, it's proportional to
$$\sum_{i = 0}^n {n\choose i} = 2^n$$
hence the mean should be
$$\frac{n\cdot 2^{n - 1}}{2^n} = \frac{n}{2}$$