# Show that $n · Y_n$ converges in distribution as n $\rightarrow 1$, and find the limit distribution.

Let $$X_1, X_2, . . .$$ be a sample from the distribution whose density is:

$$f(x) = \begin{cases} \frac{1}{2}(1+x)e^{-x}, & \text{for x>0} \\ 0, & \text{otherwise} \end{cases}$$

Set $$Y_n=min\{X_1,X_2, \dots\}$$. Show that $$n · Y_n$$ converges in distribution as n $$\rightarrow 1$$, and find the limit distribution.

I did the following:

Since, $$F(x)=1-\frac{1}{2}e^{-x}(x+2)$$

Then, $$P(n \times Y_n\le x)=P(Y_n \le \frac{x}{n})=1-(1-F(x/n))^n=1-\frac{1}{2}e^{-x/n}(2+\frac{x}{n})$$

And I don't really see where this function goes once n $$\rightarrow \infty$$

Any help would be appreciated.

## 1 Answer

You have made a mistake in your calculation. What you get is $$\lim \{1-[e^{-x/n}(1+\frac x {2n})]^{n}\}=1-e^{-x/2}$$ for $$x>0$$. Hence the limiting distribution is exponential with parameter $$\frac 1 2$$.