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An introduction to discrete random variables, their definitions, and the probability mass functions (pmf) for various types of discrete random variables, including Bernoulli, Binomial, Geometric, Negative Binomial (Pascal), and Poisson random variables. It covers the concepts of discrete-value and continuous-value random variables, the difference between them, and the pmf formulas for each discrete random variable.
Typology: Exercises
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experiments. Examples of assignments of numbers to the outcomes of
A
discrete-value (DV)
random variable has a set
occurof distinct values separated by values that cannot
be DV random variableof coin flips, card draws, dice tosses, etc... wouldA random variable associated with the outcomes
A
continuous-value (CV)
random variable may
may be finite or infinite in sizetake on any value in a continuum of values which
A DV random variable
is a
(^) Bernoulli
random variable if it
takes on only two values 0 and 1 and its pmf is
X
x
p
, x (^) = (^0)
p
x (^) = (^1)
, otherwise
and 0
p (^) <
(^) 1.
Example of a Bernoulli pmf
Binomial pmf
probability of a 1 on any single trial isIf we perform Bernoulli trials until a 1 (success) occurs and the
(^) p , the probability that the
first success will occur on the
(^) k th trial is
(^) p
1 p
k 1
. A DV random
variable
is said to be a
(^) Geometric
random variable if its pmf is
X
x
p
1 p
, otherwise
If we perform Bernoulli trials until the
(^) r th 1 occurs and the
probability of a 1 on any single trial is
(^) p , the probability that the
r th success will occur on the
(^) k th trial is
r th success on
(^) k th trial
k (^) (^1)
r (^) (^1)
p (^) r 1 p
A DV random variable
is said to be a
(^) negative - Binomial
or (^) Pascal
random variable with parameters
(^) r
and
(^) p
if its pmf is
Y
y
y (^) (^1)
r (^) (^1)
p (^) r 1 p
r , r (^) + (^) 1,
,
, otherwise
(Pascal) pmfNegative Binomial
Events occurring at random times
It can be shown that
k inside
t (^) =
(^) k
k ! n lim
n
= e
(^) k
k ! (^) e
where
t
. A DV random variable is a Poisson random
variable with parameter
if its pmf is
X
x
(^) x
x ! (^) e , x (^)
, otherwise
to another DV random variable
through
(^) g
. If the
function g is invertible, then
(^) g 1 Y
and the pmf for
is
Y
y
X
g 1
y
where P
X
x
is the pmf for X.
If the function g is not invertible the pmf and pdf of
can be found
by finding the probability of each value of
. Each value of
with
corresponding value ofnon-zero probability causes a non-zero probability for the
. So, for the
(^) i th value of
y i
x i ,^
(^) x i ,
x i ,^ n
x i , k
1
n
function.example of a non-invertible The function to the right is an