Mean and Standard Deviation of samples consisting of all $0$'s or all $1$'s

Question

Set of numbers $\ x_1, \ldots, x_m , y_1, \ldots, y_n $ where $\ x_i=0 $ for $i = 1,\ldots, m$ and $\ y_i=1 $ for $i = 1,\ldots, n$

Show that mean $M$ of this set is given by $\frac{n}{m+n}$ and the standard deviation $S$ by $\frac{ \sqrt{mn}} {m+n} $

I know the definitions of the mean and standard deviation and how to get them but Im really stuck at that question

Answer 1

The sample variance of observations denoted $x_i$ is $$S^2 = \frac{\sum_{i=1}^n (x_i - \bar x)^3}{n-1} = \frac{\sum_{i=1}^n x_i^2\; -\; (\sum_{i=1}^n x_i)^2/n}{n-1}.$$

The middle member of the equation is the definition and the last member is sometimes called the 'computational formula' (easily deduced from the definition). Is that the definition of $S$ you're using?

Notice that for $x_i \equiv 0,$ one has $\sum_{i-1}^m x_i = \sum_{i-1}^m x_i^2 = 0.$

Also, for $y_i \equiv 1,$ one has $\sum_{i-1}^n y_i = \sum_{i-1}^n y_i^2 = n.$

Put it all together, and you should be off to a good start--provided I have properly interpreted a somewhat vague question.

Answer 2

Calculate the zero, first and second moments \begin{eqnarray*} \sum 1 = ? \\ \sum z_j = ? \\ \sum z_j^2 = ? \\ \end{eqnarray*} Then use the formulea \begin{eqnarray*} \mu&=&\frac{\sum z}{\sum 1 } \\ \sigma^2 &=& \frac{\sum z^2}{\sum 1 }- \frac{(\sum z)^2}{(\sum 1)^2 }. \end{eqnarray*}

$\sum 1 = n+m \,\,\, \sum z_j = n \,\,\, \sum z_j^2 = n $