Random Variable - Random Signals and Processes Primer with MATLAB - page 151

Q.2.12. A discrete random variable is only defined in discrete points. Does this
mean that its distribution function only exists in discrete points ?
Q.2.13. Why must the PDF of a discrete random variable only have delta functions
in the discrete values of a random variable?
Q.2.14. Imagine a continuous random variable X in the interval [ a , b ] with a finite
probability of 0

< P 1. What is the probability that the random variable
X takes a particular value in the interval [ a , b ]?
Q.2.15. Does variance exist for all random variables?
Q.2.16. Is standard deviation always a measure of the width of the PDF?

2.12 Answers

A.2.1. The random variable is dimensionless. The outcomes of the experiment
can be different physical values and things. However, the values of random
variables are only numerical values without dimension (units). However,
when random signals are concerned and numerical values of a random
variable correspond to the numerical values of a physical signal, some
authors consider a random variable as a physical quantity; for example,
random voltage in electrical engineering.
A.2.2. YES. The random variable can also be defined as complex in terms of real
random variables X and Y :

Z ¼ X þ jY; (2.565)

p .
However, complex random variables are beyond the scope of this topic and
we will only considering real random variables.
A.2.3. NO. We can also label s i with a couple of numbers which lead to the
two-dimensional r.v. or n- tuple numbers leading to the n -dimensional r.v.
Additionally, we can also sometimes assign varying functions to each
outcome, leading us to the random process.
A.2.4. The domain of the random variable is the sample space S . Therefore, it would
be more correct to write X ( S ). However, since we understand that r.v is a
function of the sample space, we do not always show the dependence on S .
Instead, in most cases, we just denote random variable such as X , Y ,and Z .
A.2.5. Two following conditions must be satisfied, [PAP65, p. 88]:

where j ¼

￿ The set ( X x ) must be an event for any real number x .
￿ The probabilities of the events ( X ¼1 ) and ( X ¼1 ) are zero,

PfX ¼1g¼ 0 ; PfX ¼ 1g ¼ 0 :

(These events are not generally empty (i.e., we allow that the r.v. be equal
to 1 or 1 for some outcomes, [PAP65, p. 87])