Let X1, X2, ..., Xn, ... be a sequence of independent and identically distributed (i.i.d.) random variables having the common cumulative distribution function (cdf)
\(F(x)=\left\{\begin{array}{cc} 0, & \text { if } x<5 \\ 1-e^{5-x}, & \text { if } x \geq 5 \end{array} .\right.\)
Define Yn = min{X1, X2, ..., Xn}, Zn = √n(Yn - 5), n = 1, 2, ..., and let Z be a standard normal random variable. Then which of the following statements is true?
1
\(\displaystyle \lim _{n \rightarrow \infty} P\left(\frac{1}{2}
2
\(Y_n \stackrel{P}{\rightarrow} 5\) as n → ∞
3
\(\rm Z_n \stackrel{d}{\rightarrow} Z\) as n → ∞
4
\(\displaystyle \lim _{n \rightarrow \infty} P\left(1 = Φ(2) - Φ(1), where Φ(.) denotes the cdf of Z