Again, in this exercise how can I find the cumulative distribution function of $M_{n} - m_{n}$ and how I can calculate the limit in distribution of the sequence $(n (1 - M_{n} + m_{n})) \ n \in \mathbb{N}$?
Let $(U_{n})_{n\in\mathbb{N}}$ be a sequence of random variables i.i.d. with uniform distribution in the interval $[0, 1]$. We define for each $n\in \mathbb{N}$ the random variables: $$M_{n} = \max\{U_{1} ,...,U_{n}\}$$ $$m_{n} = \min\{U_{1} ,...,U_{n}\}$$
Show that $M_{n} -m_{n}\ \underrightarrow{d}\ 1$
Now I have to:
$$P(\min\{U_{1} ,...,U_{n}\}>a)=P(U_{1}>a)P(U_{2}>a)\dotsi P(U_{n}>a)=(1-a)^n$$ $$\rightarrow F_{m_{n}}(a)=1-(1-a)^n \rightarrow f_{m_{n}}(a)=n(1-a)^{n-1}$$
$$P(\max\{U_{1} ,...,U_{n}\}\leq a)=P(U_{1} \leq a)P(U_{2} \leq a)\dotsi P(U_{n} \leq a)=a^n$$ $$\rightarrow f_{m_{n}}(a)=na^{n-1}$$
I don't know how to continue, could you help me?
Use continuous mapping theorem:
$$m_n\xrightarrow{d}0$$
$$M_n\xrightarrow{d}1$$
Thus
$$M_n-m_n\xrightarrow{d}1$$