Question

Consider a discrete-random variable z assuming finitely many values. The cumulative distribution function, F_z(z) has the following properties:

1. \(\mathop \smallint \limits_{ - \infty }^{ + \infty } {F_z}\left( z \right)dz = 1\) 

2. F_z(z) is non-decreasing with finitely many jump-discontinuities

3. F_z(z) is negative and non-decreasing

Which of the above properties is/are correct?

1. 1 only
2. 2 only
3. 3 only
4. 2 and 3

Answer 1

Correct Answer - Option 2 : 2 only

A cumulative distribution function (CDF) is defined as:

\(P\;\left( {Z < z} \right) = \mathop \sum \limits_{ - \infty }^z f\left( z \right) = F\left( z \right)\)

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 Probabilities of discrete random variables. So CDF is always non-decreasing with finitely many jump-discontinuities.