Chapter 3: Random Variables and Probability Distributions
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CHAPTER 3 — RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS
A
random variable
is a function that associates a real number with each element of a sample
space.
•
A random variable is generally denoted by a capital letter:
X, Y,
etc.
•
An observed value of a random variable is denoted by a lowercase letter:
x, y,
etc.
Example:
Suppose a coin is flipped 3 times.
The sample space is for this experiment is
{ }
TTT
HTT
THT
TTH
THH
HTH
HHT
HHH
S
,
,
,
,
,
,
,
=
.
Letting
X
be the random variable describing the number of heads observed, we obtain:
Sample Space
Elements
x
HHH
3
HHT, HTH, THH
2
HTT, THT, TTH
1
TTT
0
Example
:
Suppose an individual plays the lottery each week until he wins the jackpot (and then
he stops).
Letting
W
represent the event that the individual wins the lottery, and letting
L
represent the event that the individual does not win, our sample space is:
{ }
K
,
,
,
,
,
LLLLW
LLLW
LLW
LW
W
S
=
We may construct a random variable,
X
, to indicate the number of weeks it takes the individual
to win the lottery:
Sample Space
Elements
x
W
1
LW
2
LLW
3
LLLW
4
LLLLW
5
M
M
In each of the above examples, the set of possible values of
X
is discrete.
Such random variables
are called
discrete random variables
.