University of Central Florida
School of Electrical Engineering and Computer Science
COP 4600  Operating Systems
Spring 2011  dcm
Probability and Statistics Concepts
Random Variable:
a rule that assigns a numerical value to each possible outcome of an ex
periment. All possible outcomes of the experiment constitute a sample space
.
A random variable
X
on a sample space
S
is a function
X
:
S
7→
R
which assigns a real
number
X
(
s
) to every sample point
s
∈
S
. This real number is called the probability of that
outcome.
A discrete random variable
maps events to values of a countable set e.g., the set of integers);
each value in the range has a probability greater than or equal to zero.
Example 1.
the experiment is a coin toss; the outcome is either
0
(head) or
1
(tail).
If the
coin is fair then
p
0
=
p
1
= 0
.
5; this means that in a large number of coin tosses we are likely
to observe heads in about half of the cases and tails in the other half of the cases. Another
example: when you throw throw a dice the outcome could be 1
,
2
,
3
,
4
,
5, or 6; for a fair dice
p
1
=
p
2
=
p
3
=
p
4
=
p
5
=
p
6
= 1
/
6.
A continuous random variable
maps events to values of an uncountable set (e.g., the real
numbers).
Example 2.
the experiment is to measure the speed of cars passing through an intersection:
the speed could be any value between 15 and 80 miles/hour. the probability of observing cars
with a speed of 19
.
1 miles/hour could be zero but the probability of observing cars with a
speed from 15 to 19
.
1 miles/hour could be
P
19
.
1
= 0
.
3 which means that 30% of the cars we
observed have a speed in the range we considered.
A discrete random variable
X
has an associated probability density function
, (also called
probability mass function)
p
X
(
x
) defined as:
p
X
(
x
) = Prob(
X
=
x
)
and a probability distribution function
also called cumulative distribution function
,
P
X
(
x
)
defined as:
P
X
(
t
) = Prob(
X
≤
t
) =
X
x
≤
t
p
X
(
x
)
Example 3.
You have a binary random variable X (the outcome is either
0
or
1
) and:
p
0
= Prob(
X
= 0) =
q
and
p
1
= Prob(
X
= 1) =
p,
with
p
+
q
= 1
.
Bernouli trials: call the outcome of 1 a “success” and ask the question what is the probability
Y
n
that in
n
Bernoulli trials we have
k
successes:
p
k
= Prob(
Y
n
=
k
) =
n
k
¶
p
k
(1

p
)
n

k
=
n
!
k
!(
n

k
)!
p
k
(1

p
)
n

k
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The binomial cumulative distribution function
is:
B
(
t
:
n, p
) =
t
X
k
=0
n
k
¶
p
k
(1

p
)
n

k
Example 4.
You have again Bernoulli trials and ask the question how many trails you need
before the first “success”.
If the first success occurs at the
i
th trial then
p
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 Spring '08
 MONTAGNE
 Normal Distribution, Operating Systems, Probability distribution, Probability theory, probability density function, Cumulative distribution function

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