Question

Suppose that the length of time Y takes a worker to complete a certain task has the probability density function given by
\(f(y) = \begin{cases} e^{-(y-\theta)}~~~if y>\theta \\0~~~elsewhere\end{cases}\)
where \(\theta\) is a positive constant that represents the minimum time until task completion. Let \(Y_1,Y_2,...,Y_n\) denote a random sample of completion times from this distribution. Find
a)The density function for \(Y_(1)=min(Y_1,Y_2,...,Y_n)\)
b) \(E(Y_(1))\)
Please, help

Answer 1

@Zarkon

Answer 2

\[f(y)=\begin{cases} e^{-(y-\theta)}~~~if ~~~y>\theta\\0~~~elsewhere \end{cases}\]

Answer 3

\(f_Y(Y_{(1)})=n (1-F_Y(y)^{n-1})f_Y(y)\)

Answer 4

closer

Answer 5

I have just that formula.

Answer 6

I don't know how to get closer. :)

Answer 7

\[f_{Y_{(1)}}(y)=n [1-F_Y(y)]^{n-1}f_Y(y)\]

Answer 8

So, what is difference between \(f_{Y_1}(y)\) and \(f_Y(y_{(1)})\)

Answer 9

one is correct notation and one is not

Answer 10

also notice how the n-1 power go to the entire quantity

Answer 11

ok. :)
My question is how to find \(F_Y(y)\)? if I take integrating of \(f_Y(y)\), what are the limits?

Answer 12

\(\theta\) to \(y\)

Answer 13

why? why is it not from theta to infinitive?

Answer 14

if you did that you would get 1

Answer 15

\[F_Y(y)=P(Y\le y)=\int_{-\infty}^{y}f_Y(t)dt\]

Answer 16

I got it.

Answer 17

if I do b), I have to apply formula \(E(Y(1))=\int(y f_{Y_1} (y)dy\)
right? what are the limits? same as part a?

Answer 18

there you integrate over the entire support

Answer 19

I don't get what you mean.

Answer 20

from theta to infinity

Answer 21

Is there anyway else or just that way?

Answer 22

i can't think of another way of the top of my head

Answer 23

We have N (normal), S (sampling), T (just call t), chi-square
what is Z,?

Answer 24

Z the random variable?

Answer 25

Yes

Answer 26

typically Z is a standard normal random variable

Answer 27

you mean Z~N(0,1)?

Answer 28

yes

Answer 29

On the formula, n, n-1 apply to find some value of distribution, but they are number of freedom, right? if n large enough, how are they different to each other?

Answer 30

n just represents the number of random variable you are considering.

Answer 31

at least for this problem

Answer 32

I am ok with this problem. Thanks for the help. I just want to make clear the notations from my lecture. :)

Answer 33

did you find the expected value?

Answer 34

Sometimes, my Prof used U, V, Z, W, T, S..... some of them are universal notation, some are just the name of a particular problem. I got confused.

Answer 35

ah...ic

Answer 36

I didn't find it yet.

Answer 37

ok...I did...if you get an answer let me know

Answer 38

I know you are a statistic Prof so that I try to ask my unclear information first. :)

Answer 39

Yes, I will. Again, thanks for the help

Answer 40

I am sorry. I was on the phone. This is my work

Answer 41

good

Answer 42

:)