SOLUTION: Five observations have been drawn independently and at random from
a continuous distribution. What is the probability that the next observation
will be less than all of the first
Algebra.Com
Question 1164707: Five observations have been drawn independently and at random from
a continuous distribution. What is the probability that the next observation
will be less than all of the first five?
Found 2 solutions by CPhill, greenestamps:
Answer by CPhill(2189) (Show Source): You can put this solution on YOUR website!
This is a classic problem in **non-parametric statistics** and **symmetry**. Because the observations are drawn from a continuous distribution, the probability that any two observations are exactly equal is zero.
### The Symmetry Argument
When you have a set of independent and identically distributed (i.i.d.) random variables, every possible ordering of those variables is **equally likely**.
1. Let the first five observations be .
2. Let the next observation be .
3. We now have a total of **6 observations** in our set.
In a set of 6 independent observations from the same continuous distribution, any one of them is just as likely to be the "minimum" as any other.
### Calculation
Since there are 6 observations and each has an equal chance of being the smallest value:
* The probability that is the smallest is .
* The probability that is the smallest is .
* ...and so on.
The probability that the **next observation ()** is the smallest (less than all of the first five) is:
**The probability is 1/6 (or approximately 0.1667).**
---
### Key Intuition
It does not matter what the actual distribution is (Normal, Uniform, Exponential, etc.). As long as the observations are independent and come from the same continuous distribution, the "rank" of the sixth observation is distributed uniformly among the possibilities .
Would you like to see how this probability changes if we were looking for the probability that the sixth observation is the *median* instead of the *minimum*?
Answer by greenestamps(13295) (Show Source): You can put this solution on YOUR website!
The problem is equivalent to this:
Make 6 observations and list them in order from least to greatest. What is the probability that the 6th observation is first in the list?
Using the basic definition of probability, there are 6 possibilities for which one of the 6 observations is first; there is only one possibility for the 6th observation to be first; so the probability that the 6th observation is first in the list is 1/6.
ANSWER: 1/6
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