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Math 3338: Probability
Discrete Probability Distributions
Pankaj Singh
Department of Mathematics, University of Houston
[email protected]
math.uh.edu/∼pankaj/math3338
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Probability Space
A Probability Space is a triple (S, E, P), where S is the sample space, E
is the event space, and P is a real valued function having the following
properties:
0 ≤ P(E ) ≤ 1 for all E ∈ E;
P(S) = 1;
P
∪∞
j=1 Ej
=
∞
X
P(Ej )
j=1
for any sequence of events E1 , E2 , · · · with the property that
Ei ∩ Ej = ∅ whenever i 6= j (sequence of mutually exclusive events).
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Random Variables and Their Probability Distributions
A random variable for a probability space (S, E, P) is a real-valued
function whose domain is a sample space.
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Example
Let X represent the number of heads obtained from tossing a fair coin 3
times.
List the possible values of X.
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Discrete Random Variable
A random variable X is said to be discrete if it can take on only a finite
number - or a countably infinite number -of possible values x. The
probability mass function of X , denoted by p(x), assigns probability to
each value x of X so that the following conditions are satisfied:
P(X = x) = p(x) ≥ 0.
X
P(X = x) = 1, where the sum is over all possible values of x.
x
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Determine the probability mass function for X.
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Cumulative Distribution Function (CDF).
The cumulative distribution function F(b) for a random variable X is
F (b) = P(X ≤ b)
if X is discrete,
F (b) =
∞
X
p(x),
x=−∞
where p(x) is the probability function. The cumulative distribution
function (CDF) is sometimes also called the distribution function. The
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Determine the cumulative distribution function for X .
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Sketch the graph of the cumulative distribution function for X .
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Properties of CDF
The cumulative distribution function for a random variable X is a function
FX : R → [0, 1] with the following properties:
FX (x) = P(X ≤ x)
lim FX (x + ) = FX (x) for all x
→0+
lim FX (x) = 0
x→−∞
lim FX (x) = 1
x→∞
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

0,





0.05,





0.15,
Suppose FX (x) = 0.35,



0.65,





0.85,




1,
if
if
if
if
if
if
if
x <1
1≤x <2
2≤x <3
3≤x <4
4≤x <5
5≤x <6
x ≥6
Graph the cumulative distribution function of X .
Find the probability mass function of X .
Graph the probability mass function of X .
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