1) A box contains M disks, numbered from 1 to M. Disks are drawn from the box independently without replacement. What is the probability of the nth disk drawn shows a higher number than any of the previous (n-1) disk?
Let X_n be a random variable taking value 1 if the nth disk shows a highest number than any of the previous (n-1) disk , and 0 otherwise.
I have no clue how to do this question. Please help me some guidance how to do it, I mean what kind of distribution model I should use?
2) Let the sequence X_n = X_1,X_2,....,X_M marks the times at which records( highest observed values) are broken (exceeded)
Let Y(m,n) be the number of times a record is broken between times m and n ( inclusive). Write down an expression for Y(m,n) in term of the Xs and hence derive and expression for the expected number of records betweem times m and n
Question 2 is related from part 1. So, in order to do this I think we have to use the result from part 1 but apparently I dont know how to do part 1, so I got stuck on question 2.
I hope some one could give me some hints.
Thank you for your time
How can disks be drawn from a box independently without replacement?
For example, say you have disks numbered 1 through 10. If you first pick disk 4, and you do not replace another (or the same) disk 4 into the box, then your chances of drawing a 7, for example, has gone from 1/10 to 1/9. Therefore, we have a dependent relationship.