7/23/2019 Probability Problem Set Question 2 Sol

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What is the chance that X 

1  is the smallest of the ve draws?

A sample of 5 will be drawn from the same continuous distribution. Since thesenumbers will be drawn independently, does the second number drawn (value of  X 

2¿

 have more chance of being the lowest compared to the rst number

(value of   X 

1¿

? No.

Any of these numbers could be the least with eual probability because of

independence. !ence, P ( X 1=min { X 1 , X 2, X 3 , X 4 , X 5 })=

1

5.

What is the chance that the ve draws are made in increasing order?

Since distribution being drawn from is continuous, the probability of any "numbers being identical is #ero. So, the sample of 5 would always be 5 distinctnumbers.

Since these numbers will be distinct, any one of them could be the lowest, thesecond lowest, and so on. $e %now from above that any of these 5 numberscould be the lowest one with eual probability. Now, remove that number. $ehave & numbers remaining. Again, any one of these & could be the lowestamongst these & with eual probability. So, the second lowest number, amongstthe ve, could be any of them with eual probability.

So, basically, any order can emerge with eual probability. !ow many orders are

there in total? 5')"*. +n how many order the ve draws are made in increasingorder? ust ), where

 X 1  is the lowest,

 X 2  is the second lowest, - ,

 X 5  is

the highest.

!ence, the answer is1

120.