STA3032, Chapters 5 & 6, Page 1 of 8
Revised on August 22, 2011
Chapters 5 and 6
Some Commonly Used Distributions
Sections 5.1, 5.2, 6.1 and 6.2 Random Variables
Definition:
A
random variable
is a function that assigns numerical value to each outcome in a
sample space. Random variables are usually denoted by capital letters that are at the end of the
alphabet.
Quantitative random variables can be either
discrete
or
continuous.
A random variable
is said to
be discrete if its values can be listed. A continuous random variable has values within an interval
of the real line, or a union of intervals or the whole real line.
Comparison of the discrete and continuous random variables
Properties
Discrete
Continuous
Possible values
Can be listed
Infinitely many
Probability distribution
p(x) = P(X=x) (pmf)
f(x) (pdf)
Conditions for function
0 ≤ p(x) ≤ 1
&
( ) 1
all x
px
f(x) ≥ 0
&
( )
1
f x dx
Cum. Dist. function (cdf)
( )
(
)
( )
tx
F x
P X
x
p t
.
( )
(
)
( )
x
F x
P X
x
f t dt
.
E(X) = Mean (μ)
()
X
all x
xp x
X
xf x dx
Variance (σ
2
)
22
(
)
( )
XX
all x
x
p x
(
)
( )
x
f x dx
Standard Deviation (σ)
2
2
Properties of the cdf:
1.
Nondecreasing with 0 ≤ F(x) ≤ 1.
2.
lim
( )
0
lim
( ) 1.
xx
F x
and
F x
3.
Discrete rvs have F(x) that is a stepfunction.
4.
Continuous, except at, at most a finite number of points
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Empirical Rule:
For a mound shaped, somewhat symmetric distribution of a random variable X,
with mean μ
X
and standard deviation σ
X
,
1
0.68
2
0.95
3
1.00
XX
PX
Chebyshev’s Inequality
Let X be a random variable with mean μ and standard deviation σ. Then for any k > 1,
2
1
1
P X
k
k
. Equivalently
2
1
P X
k
k
Sections 5.3
to 5.7 The Binomial Family of distributions
The binomial family includes the Bernoulli distribution, Binomial distribution, Geometric
distribution,
Negative
Binomial
distribution,
Poisson
distribution
and
hypergeometric
distribution. In all but the last one, we assume that there is an experiment with two
possible
outcomes,
labeled “Success” (or S = the outcome we are interested in and count) and “Failure”
(or F)
You should notice that these distributions are all interrelated. There are similarities and
differences between them. The differences define when you should use each one. Here is a list of
these distributions with their similarities and differences:
Distributions and Conditions when they are used:
In each of the following distributions, we are assuming that a
Bernoulli experiment
(with only
two possible outcomes, S and F) is repeated.
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 Summer '08
 Kyung
 Statistics, Normal Distribution, Probability theory, π

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