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CIS 2033 based on
Dekking et al. A Modern Introduction to Probability
and Statistics. 2007
Longin Jan Latecki
Discrete Random Variables
Discrete random variables are obtained by counting and have sample spaces which
Are countable. The values that represent each outcome are usually integers.
Random variables are denoted by capital letters.
X: is the number of times that we flip a coin until H comes up
The possible outcomes are denoted by lower case letters:
a=1, a=2, a=3...
Probability Mass Function
The probability mass function (pmf) of a discrete random variable maps each possible
outcome in the sample space to it's corresponding probability.
The sum of the probabilities of all possible outcomes will always be equal to 1.
Probability Distribution Function
The distribution function of a random variable X, also referred to as the
cumulative distribution function (CDF) yields the probability
that X will take a value less than or equal to a.
Hence, the value of F(a) is equal to the sum of all probabilities
of outcomes less than or equal to a:
F (a)   P( X  ai )  P( X  a )  P( X  a )
ai a
If we are given a CDF, we can get the pmf with the following formula:
P( X  a )  P( X  a )  P( X  a )  P( X  a )  P( X  a   )  F ( a )  F ( a   )
for some sufficiently small ε>0.
Graphs of pmf and CDF
Cumulative Distribution Function
Probability Distribution Function
F (a)   P( X  ai )  P( X  a )  P( X  a )
ai a
Bernoulli Distribution
The Bernoulli distribution is used to model an
experiment with only two outcomes, success
and failure. The parameter p is the chance for
An example is flipping a coin, where “heads”
may be success and “tails” may be failure.
Binomial Distribution
The Binomial Distribution represents multiple
Bernoulli trials. The parameter n is the number of
trials, and the parameter p is the probability of
success as in the Bernoulli distribution.
P(X=k) is the probability of k successful outcomes in
n trials.
Geometric Distribution
A geometric distribution gives information about the
probability of success after k attempts. The parameter
p is the probability of success on the kth try.
This means that all previous k-1 tries failed
An example of this would be finding the probability that
you will hit a bullseye with a dart on your kth toss.