# Distribution of sum of Indicator Variables

Let $$X1$$, $$X2$$, $$X3$$...$$Xn$$ be n observations with distribution function $$F$$. Let $$F^{*}$$ be the empirical distribution of the random sample.

$$F^{*} = \frac{1}{n} \sum I(X_{i} \le x)$$ where I = 1, if the equality holds, 0 otherwise.

i) Find the distribution of $$\sum I(X_{i} \le x)$$

ii) Find the limiting distribution of $${\sqrt(n)} [F^{*}(x)-F(x)]$$

This is my approach;

Let r observations satisfy the equality and (n-r) observations don't.

$$F^{*} = \frac{1}{n} (1+1+..1+0+0+..+0) = \frac{r}{n}$$

$$\sum I(X_{i} \le x) = r$$

So distribution is the number of successes in the sample.

But I'm not sure if this is the way to approach the problem.

For the 2nd part;

$$Y = {\sqrt(n)} [F^{*}(x)-F(x)] = {\sqrt(n)} [r/n - p] = \frac{1}{\sqrt(n)} [r-np]$$

$$n \rightarrow \infty , r\rightarrow np$$

therefore,

$$(r-np) \sim N(0, \sigma^{2})$$

$$Y = \frac{1}{\sqrt(n)} [r-np] \sim N(0, \frac{\sigma^{2}}{n})$$

• Hint. The $X_i$ have to be independent for the problem to have meaning. Assuming that is so, then $\Sigma I$ is binomial distributed (you figure out n and p) and (ii) is easily obtained from the CLT for the binomial. – Gordon Smyth Jun 24 at 7:47
• @GordonSmyth Is p = F(x) . But should I relate that to r? – Dom Jo Jun 24 at 8:02
• @GordonSmyth I have updated it. Please check – Dom Jo Jun 24 at 8:34
• You haven't yet defined what $\sigma^2$ is. – Gordon Smyth Jun 24 at 11:38
• @GordonSmyth that would be the variance of binomial RV = npq correct? – Dom Jo Jun 24 at 12:33