UNR, MATH/STAT 352, Spring 2007
Random variable (rv) –
central concept of probability theory
Numerical description of experiment
UNR, MATH/STAT 352, Spring 2007
Sample space
S
Definition:
A random variable X is a function
whose domain is the sample space
and range is the real line.
Outcome
x
UNR, MATH/STAT 352, Spring 2007
Experiment: tossing a die
Sample space
Probabilities
1/6
1/6
1/6
Random variable
X = face value
UNR, MATH/STAT 352, Spring 2007
1/6
1/6
1/6
Experiment: tossing a die
Sample space
Probabilities
1/6
1/6
1/6
1/6
1/6
Random variable
X = 1 for odd, 0 for even
face value
P(X=1) = P(X=0) = 1/2
UNR, MATH/STAT 352, Spring 2007
1/6
Probabilities
Tail
Sample space
Head
Experiment: tossing a coin
1/2
1/2
Random variable
X = 1 for T, 0 for H
P(X=1) = P(X=0) = 1/2
UNR, MATH/STAT 352, Spring 2007
P(X=1) = P(X=0) = 1/2
UNR, MATH/STAT 352, Spring 2007
Experiment: tossing two dice
Second die
Sample space
(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)
First die
(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)
(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)
Each outcome (n,m)
has probability
1/36
(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)
(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)
(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)
UNR, MATH/STAT 352, Spring 2007
Experiment: tossing two dice
X = sum of faces
Sample space
(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)
2
3
4
5
6
7
(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)
3
4
5
6
7
8
(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)
4
5
6
7
8
9
(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)
5
6
7
8
9
10
(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)
6
7
8
9
10
11
(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)
7
8
9
10
11
12
UNR, MATH/STAT 352, Spring 2007
Experiment: tossing two dice
X = sum of faces
Possible values
2 3 4 5 6 7 8 9 10 11 12
1
2
3
4
5
6
5
4
3
2
1
36 36 36 36 36 36 36 36 36 36 36
Probabilities
UNR, MATH/STAT 352, Spring 2007
2
3
4
5
6
7
3
4
5
6
7
8
4
5
6
7
8
9
5
6
7
8
9
10
6
7
8
9
10
11
7
8
9
10
11
12
Old car
http://www.geocities.com/aruba692003/Old-car.jpg
+
+
http://www.autoinspection.com/back/aselogo.gif
Experiment: old car with four cylinder engine goes to inspection
http://www.seanhylandmotorsport.com/frppcrate/4cyleng.jpg
Four-cylinder engine
X = number of cylinders with low compression
X = {0, 1, 2, 3, 4}
UNR, MATH/STAT 352, Spring 2007
Experiment: measuring the life time of a device (computer chip)
Failure
Time
X = time until failure
X>0
UNR, MATH/STAT 352, Spring 2007
The same sample space may be associated
with different random variables
In general, it is impossible to reconstruct the original
sample space that was used to construct a random variable
from the knowledge of this random variable
We work with random variables and do not think of
original sample spaces
There are differences in working with discrete and
continuous random variables
UNR, MATH/STAT 352, Spring 2007