Consider there is a discrete random process x(t) which includes N number of samples. x(t) crosses the level L n number of times. The level crossing probability of x(t) with level L will then be:
- P1=n/(N-1).
However I need to use a probability which is defined by dividing n with N (instead of N-1):
- P2=n/N.
Is there a name for such probability (P2)?
If we talk about variances the used terms are unbiased and biased variance, however I need to know how to call the probability P2.
Since n samples out of N samples cross the level L, then by definition the probability to find a sample with a level greater than L will be = n/N.
Thank you for your answer Abdelhalim Zekry, however i am a little confused about the proper definition. From my point of view it's not the samples that cross a level but the transition between two samples. Let's look at it this way. If there is only 1 sample the crossing probability for any level is zero. You need at least N=2 samples and that makes N-1 transitions or N-1 potential crossings.
Dear Boris,
If you have one ample and you want to determine the probability of crossing specific level, then you have to define the levels and their number as occurring with the level quantification. If you have Q levels. then the probability to that the sample has one specific level is 1/Q.
On the other side.the statistical probability is thought for large ensembles not for small numbers. You have also to speak about level ranges or intervals of levels.
Lastly do you speak from probability of transition or probability to exist higher certain level L? You need to count all transitions say N and the transitions to a level higher than the level L say n and then you get the probability of the transition to a level higher than L by dividing n by N.
wish you the best
Dear Adbelhalim,
Yes i mean probability of transition, not probability to exist above certain level.
I was able to sidestep the problem by evaluating P2 as a function of P1:
- P2=n/N=P1.(N-1)/N=P1.(1-1/N)
Thank you for your time and help.
hi
Your q is interesting. It seems like a Markov process. Pl see if following (theoretical probability) is of help.
1 Consider number of levels (sample space ) = Lt
2 Probability of any level (existing before expt begins) is with uniform pmf Pe1 =1/L1
3. Now all possible states s1 s2... are L1 in number and probability of any one state sk is Pe1 =1/L1
4.Next if it is in state sk (assume uniform transition probability) it can transit to any of L1 states (including itself). But we want to identify those levels which are above some level > sw. As the sk also identifies levels we can say (uniform pmf) qtrans =( L1-sk)/L1.
5 Now probability of occurrence level sw and transiting to higher is = (sw/L1)*( L1-sk)/L1.
6. Probability is tricky 5 is valid provided we are blind to first event. in case we "see" it then we need to consider only the transition probability qtrans= ( L1-sk)/L1..(this is conditional probability P(x|y) ).
Cheers
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