Question

Assure thatha lenale randon Sample om epop iem s papulalion widh the Salouin pralshility S>, a where-a-26 Find the and 02上。

Answer 1

$The~cdf~of~n\beta X_{(1)}~is\\ F_n(x)=P(n\beta X_{(1)}\leq x)=1-P\left(X_{(1)}>\frac{x}{n\beta} \right )\\ =1-P(X_1>\frac{x}{n\beta},...,X_n>\frac{x}{n\beta})=1-\prod_{i=1}^nP(X_i>\frac{x}{n\beta})~since~X_i's~are~independent\\ =1-\prod_{i=1}^n\left(\int_{\frac{x}{n\beta}}^\infty \frac{\alpha\beta}{(\alpha+\beta x)^2} dx\right )=1-\prod_{i=1}^n\left(\int_{\alpha+\frac{x}{n}}^\infty \frac{\alpha}{t^2} dt\right ),~take,~\alpha+\beta x=t\\ =1+\prod_{i=1}^n\left[\frac{\alpha}{t} \right ]^\infty_{\alpha+\frac{x}{n}}=1-\left(1+ \frac{x}{n\alpha}\right)^{-n}\rightarrow 1-\exp\left(-\frac{x}{\alpha} \right )~as~n\rightarrow \infty\\ which~is~cdf~of~exponential~distribution~with~mean=\alpha.$

$Hence~n\beta X_{(1)}\overset{L}{\rightarrow}X\sim Exponential~distribution~with~mean=\alpha.$