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Chapter 5.Random Variables.
• A random variable X associates a numerical value with
each outcome of an experiment.
(It is frequently the case when an experiment is performed that we
are mainly interested in some function of the outcome as opposed to
the actual outcome itself).
•
Examples:
Examples:
Experiment
Sample space
Event
Random
Experiment
Sample space
Event
Random
Variable
Variable
1)
1)
Coin toss
S={ H,T}
A=
“H”
let X
Coin toss
S={ H,T}
A=
“H”
let X
# of heads
,
# of heads
,
H(A) = 1
,
possible values
for X :
x=0,1
H(A) = 1
,
for X :
2)
2)
Two
Two
dice toss
,S= {(1,1)(1,2)…….
.}
Let X
dice toss
,S= {(1,1)(1,2)…….
.}
Let X
 the two dice total
the two dice total
X((1,1))=1+1 =2
x= 2,3,……12
x= 2,3,……12
3) Testing 8 elderly adult for the allergic reaction
(yes or n
3) Testing 8 elderly adult for the allergic reaction
(yes or n
o)
o)
S we have
256 possible outcomes ,for instance
:
we have
256 possible outcomes ,for instance
:
A=(yes,no,yes,no,no,no,yes,no)
.
A=(yes,no,yes,no,no,no,yes,no)
.
Let X
Let X
 # of allergic reactions among set of eight adults .
# of allergic reactions among set of eight adults .
x=0,1,2,3,4,5,6,7,8
X(A)=3
X(A)=3
•
Random variables can be
discrete (if it has either a
finite number of values or infinitely many values
finite number of values or infinitely many values
that can be arranged in a sequence) ,
that can be arranged in a sequence) ,
or
continuous(measurements on a continuous scale).
continuous(measurements on a continuous scale).
•
The
probability distribution for a discrete random
probability distribution for a discrete random
variable
variable
X
X
is a graph, table or formula that gives the
possible values of
x
and the probability
f(x)=P(X=x)
associated with each value.
•
Properties of a probability distribution:
1) 0
≤
f(x
i
)
≤
1;
2) 2)
∑
f(x
i
)
=1.
Example 1
• Toss a fair coin three times and
define X
= number of heads.
1/8
1/8
1/8
1/8
1/8
1/8
1/8
1/8
P(X
=
0) =
1/8
P(
X =
1) =
3/8
P(
X =
2) =
3/8
P(
X =
3) =
1/8
HHH
HHT
HTH
THH
HTT
THT
TTH
TTT
X
3
2
2
2
1
1
1
0
1/8
3
3/8
2
3/8
1
1/8
0
f(x)
x
Probability
Histogram for
X
Cumulative Distribution Function
• What is the probability that at least one is a subscriber?
• Can use the
cumulative distribution function
: a
function that specify, for each value x, the probability that
X
≤
x.
• What is the probability that at most 1 is a subscriber?
F(x)=P(X
≤
x) = p(X=x
1
)+p(X=x
2
)+….p(X=x
k
)
with x
k
≤
x < x
k+1
X
f(x)
0
.49
1
.42
2
.09
Exercise 1. Let X is the number of subscribers
to a magazine in a sample of 2
X
f(x)
F(x)
0
.49
.49
1
.42
.91
2
.09
1.00
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Exercise 2
From the six marbles numbered :
1,1,1,1,2,2
Two marbles will be drawn at random without
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This note was uploaded on 04/17/2008 for the course HONORS 191 taught by Professor Palmero during the Spring '08 term at UMass (Amherst).
 Spring '08
 palmero

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