1
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:
Experiment
Sample space
Event
Random Variable
1) Coin toss
S={ H,T}
A=
“H”
let X -# of heads
,
X(A) = 1
,
possible values
for X :
x=0,1
2) Two-dice toss , S= {(1,1)(1,2)…….
.}
Let X- be the two dice total
X((1,1))=1+1 =2
,
x= 2,3,……12
3) Testing 8 elderly adults for the allergic reaction
(yes or no)
S- we have
256 possible outcomes ,for instance
the event
A=(yes,no,yes,no,no,no,yes,no)
.
Let X- # of allergic reactions among the set of eight adults .
x=0,1,2,3,4,5,6,7,8
X(A)=3
•
Random variables can be
discrete (if it has either a
finite number of values or infinitely many values
that can be arranged in a sequence) ,
or
continuous(measurements on a continuous scale).
•
The
probability distribution for a discrete random
variable
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
a) What is the probability that at least one is a subscriber?
b) What is the probability that at most 1 is a subscriber?
We can use the
cumulative distribution function
: a function that
specifies, for each value x, the probability that X
≤
x.
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