Question
Download Solution PDFThe probability cumulative distribution must be monotone and
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFA cumulative distribution function (CDF) is defined as:
which is the probability that Z is less than or equal to some specific z, i.e. it defines a cumulative sum of up to a specified value of z.
Also, f(z) is the probability density function.
Since the probability is always ≥ 0.
So, CDF is the summation of the Probabilities of discrete random variables. So CDF is always non-decreasing with finitely many jump-discontinuities.
Mathematically it is defined as FZ[zi] = P[Z ≤ zi]
Consider the example of dice. Let, X is the random variable.
| X | P[X=x] |
| 1 | 1/6 |
| 2 | 2/6 |
| 3 | 3/6 |
| 4 | 4/6 |
| 5 | 5/6 |
| 6 | 6/6 |
The distribution function is shown below
Last updated on Jul 2, 2025
-> ESE Mains 2025 exam date has been released. As per the schedule, UPSC IES Mains exam 2025 will be conducted on August 10.
-> UPSC ESE result 2025 has been released. Candidates can download the ESE prelims result PDF from here.
-> UPSC ESE admit card 2025 for the prelims exam has been released.
-> The UPSC IES Prelims 2025 will be held on 8th June 2025.
-> The selection process includes a Prelims and a Mains Examination, followed by a Personality Test/Interview.
-> Candidates should attempt the UPSC IES mock tests to increase their efficiency. The UPSC IES previous year papers can be downloaded here.