RANDOM
VARIABLES
Experiment
An Experiment is an activity
involving chance that can
be have different results.
Flipping a coin and tolling a
number cube are examples
of experiment.
Event
Consist of one or more
outcomes of a probability
experiment.
P(event)=number of times
t
the event occur
n
number of trials
Cumulative distribution
function
The cumulative distribution
function (cdf), denoted by
F(x),
measures
the
probability that the random
variable X assumes a value
less than or equal to x,
That is F(x)=P(X<=x)
Random process
Sample space
Collection
of
time
functions
(signals)
corresponding
to
various outcomes of
random experiments.
Example: channel noise
Interfernce
The sample space of an
experiment is the set of
all possible outcomes.
You can use { } to show
sample space.
Example: When a coin is
flipped, {head, tails} is
the sample space.
Definition RV
A random variable is a
set of possible values
from
a
random
experiment.
There are two types of
RV.
discrete RV
continuous RV
Distribution
A probability distribution
is the set of values that
could
occur
for
a
random
variable,
together with the longrun relative frequencies
with which they do occur
when
that
random
variable
is
actually
observed.
Binomial random
variable
Hypergeometric
distribution
Binomial
random
variable is a random
variable that describe
the outcomes of a
binomial experiment.
Example: Rolling a fair
die 50 times and
record the numbers.
In
probability,
the
hypergeometric
distribution is a discrete
probability distribution
that
describes
the
probability of success in
draws,
without
replacement.
"Bernoulli trials" - experiments satisfying 3 conditions:
1. Experiment has only 2 possible outcomes: Success, S and Failure, F.
2.The probability of S is fixed (does not change) from trial to trial.
P(S)=p, 0<p<1, P(F)=1-P(S)=1-p.
3. n independent trials of the experiment are performed.
More about discrete random variable
Probability notation for a discrete random variable
The following notation specify a probability for a
possible outcome of a discrete random variable
X=the random variable
k=a specific number the discrete random variable
could assume
P(X=k) is the probability that X equals k.
The function of random variable
Element or Members
where each element is a
possible outcome.
Example: {a,b,c,d} | a is
an element.
Probability mass
function
Probability
mass/density functions
are used to describe
discrete and continuous
probability distributions,
respectively.
Discrete uniform
distribution
The discrete uniform
distribution
occurs
when these are a finite
number (m) of equally
likely
outcomes
possible.
Cumulative distribution
function
Bernoulli distribution
A cumulative distribution
function is a table or rule that
gives P(X<=k) for any real
number k.
More about continuous random variable
For a continuous random variable, we are only
able to find probabilities for intervals of values.
Notation for probability in an interval
The two endpoints of an interval are represented
using the letters a and b.
The interval of values of X that fall between a and
b, including the two endpoints, is written
a<=X<=b.
The probability that X has a value between a and
b is written P(a<=X<=b)
if X is a random variable and Y=g(X), then Y itself is a
random variable. Thus, we can talk about its PMF, CDF, and
expected value. First, note that the range of Y can be
written as
RY={g(x)|x RX}.
If we already know the PMF of X, to find the PMF of Y=g(X), we can write
PY(y)=P(Y=y)=P(g(X)=y)=∑x:g(x)=yPX(x)
∈
Elements