- #1
DenisaZsuZsa
- 2
- 0
Homework Statement
You have an urn that contains n balls labeled with the natural numbers {1,2,3,...,n} and you extract n balls from the urn (with the condition that a ball may not be returned to the urn once drawn). You have to determine the distribution of X=(X1,X2,...,Xn), where Xk is the number of the ball from the k-th extraction.
Homework Equations
For Z a discrete random variable, that can take the values 1, ..., n
Probability Distribution
p(z)=P(Z=z)
Cumulative Distribution Function
F(z)=P(Z<z)
where z is in {1,...,n}
The Attempt at a Solution
There are n! possible cases.
Xk (k={1,...,n}) are discrete random variable. Before the k-th extraction in the urn there are n-k+1 balls left with the probability of occurrence = 1/(n-k+1) and Xk are discrete random variable:
1 2 ... n
X1: ( 1/n 1/n ... 1/n )considering i1 the number of the ball extracted on the first extraction, we have:
1 ... i1-1 i1 i1+1 .. n
X2: ( 1/(n-1) ... 1/(n-1) 0 1/(n-1) .. 1/(n-1) )that means that for i1 the probability of occurrence = 0.
...
before the last extraction there is 1 ball left in urn, that obviously, has the probability of occurrence = 1.
I have no clue what to do next and I need it to demonstrate something for a bigger project. I do not ask you for the solution, I just need some hints.
P.S.: I hope you will understand what I wrote here, I've tried my best. Sorry if you don't, english is not my native language.