A random variable, usually written X, is a variable whose possible values are numerical outcomes of a random phenomenon.

**RANDOM VARIABLES:**

A random
variable, usually written X, is a variable whose possible values are numerical
outcomes of a random phenomenon. Random variable consists of two types they are
discrete and continuous type variable this defines discrete- or continuous-time
random processes. Sample function values may take on discrete or continuous a
value is defines discrete- or continuous Sample
function values may take on discrete or continuous values. This defines
discrete- or continuous-parameter random process.

^{ü }**RANDOM PROCESSES VS. RANDOM VARIABLES:**^{}

• For a
random variable, the outcome of a random experiment is mapped onto variable,
e.g., a number.

• For a
random processes, the outcome of a random experiment is mapped onto a waveform
that is a function of time.Suppose that we observe a random process X(t) at
some time t1 to generate the servation X(t1) and that the number of possible
waveforms is finite. If Xi(t1) is observed with probability Pi, the collection
of numbers {Xi(t1)}, i =1, 2, . . . , n forms a random variable, denoted by X(t1),
having the probability distribution Pi, i = 1, 2, . . . , n. E[ ・ ] =
ensemble average operator.

^{ü} **DISCRETE RANDOM VARIABLES:**

A
discrete random variable is one which may take on only a countable number of
distinct values such as 0,1,2,3,4,........ Discrete random variables are
usually (but not necessarily) counts. If a random variable can take only a
finite number of distinct values, then it must be discrete. Examples of
discrete random variables include the number of children in a family, the
Friday night attendance at a cinema, the number of patients in a doctor's surgery,
the number of defective light bulbs in a box of ten.

^{ü} **PROBABILITY DISTRIBUTION:**

The
probability distribution of a discrete random variable is a list of
probabilities associated with each of its possible values. It is also sometimes
called the probability function or the probability mass function. Suppose a
random variable X may take k different values, with the probability that X = x_{i}
defined to be P(X = x_{i}) = p_{i}. The probabilities p_{i}
must satisfy the following:

1: 0 < p_{i}
< 1 for each i

2: p_{1}
+ p_{2} + ... + p_{k} = 1.

All
random variables (discrete and continuous) have a cumulative distribution
function. It is a function giving the probability that the random variable X is
less than or equal to x, for every value x. For a discrete random variable, the
cumulative distribution function is found by summing up the probabilities.

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Communication Theory : Random Process : Random Variables |

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