Let Y1 be the smallest observation of three independent

Chapter 3, Problem 3E

(choose chapter or problem)

QUESTION:

Let $Y_{1}$ be the smallest observation of three independent random variables $W_{1}, W_{2}, W_{3}$, each with a Weibull distribution with parameters $\alpha$ and $\beta$. Show that $Y_{1}$ has a Weibull distribution. What are the parameters of this latter distribution? HINT:

\begin{aligned} G\left(y_{1}\right) &=P\left(Y_{1} \leq y_{1}\right)=1-P\left(y_{1}<W_{i}, i=1,2,3\right) \\ &=1-\left[P\left(y_{1}<W_{1}\right)\right]^{3} .\end{aligned}\)

Equation Transcription:

${Y}_{1}$

${W}_{1},{W}_{2},{W}_{3}$

$\alpha{}$

$\beta{}$

${Y}_{1}$

$G\left({{y}_{1}}\right)\ \ =P\left({{Y}_{1}\leq{}{y}_{1}}\right)=1-P\left({{y}_{1}<{W}_{i},i=1,2,3}\right)\ \ \ =1-{\left[{P\left({{y}_{1}<{W}_{1}}\right)}\right]}^{3}\$

Text Transcription:

Y_1

W_1,W_2,W_3

Alpha

Beta

G(y_1) =P(Y_1 < or = y_1)=1-P(y_1<W_i, i=1,2,3) =1-[P(y_1<W_1)]^3

Questions & Answers

QUESTION:

\begin{aligned} G\left(y_{1}\right) &=P\left(Y_{1} \leq y_{1}\right)=1-P\left(y_{1}<W_{i}, i=1,2,3\right) \\ &=1-\left[P\left(y_{1}<W_{1}\right)\right]^{3} .\end{aligned}\)

Equation Transcription:

${Y}_{1}$

${W}_{1},{W}_{2},{W}_{3}$

$\alpha{}$

$\beta{}$

${Y}_{1}$

$G\left({{y}_{1}}\right)\ \ =P\left({{Y}_{1}\leq{}{y}_{1}}\right)=1-P\left({{y}_{1}<{W}_{i},i=1,2,3}\right)\ \ \ =1-{\left[{P\left({{y}_{1}<{W}_{1}}\right)}\right]}^{3}\$

Text Transcription:

Y_1

W_1,W_2,W_3

Alpha

Beta

G(y_1) =P(Y_1 < or = y_1)=1-P(y_1<W_i, i=1,2,3) =1-[P(y_1<W_1)]^3

ANSWER:

Answer:

Step 1 of 1 :

Let ${Y}_{1}$ be the smallest observation and ${W}_{1}$ , ${W}_{2}$ and ${W}_{3}$ are the independent random variable.