Download Discrete Random Variables

Quantitative Methods II 360-255-LW
Vincent Carrier
Discrete Random Variables
A random variable X is a variable whose value depends on the outcome of a random
experiment.
Let X be a random variable taking values in some set DX . If DX only contains natural
numbers, then we say that X is a discrete random variable.
Example: The following random variables X are discrete random variables.
a) X : number obtained by throwing a die
DX = {1, 2, 3, 4, 5, 6}
b) X : number of throws of a coin to get a head
DX = {1, 2, 3, 4, . . .}
c) X : number of plane crashes in a year
DX = {0, 1, 2, 3, 4, . . .}
It is customary to distinguish between the random variable X with a capital letter and
its observed value x with a small letter.
To each random variable X is associated a probability distribution p : DX → [0, 1]
such that
p(x) = P (X = x)
for x ∈ DX .
A probability distribution always satisfies the conditions
1) 0 ≤ p(x) ≤ 1
for x ∈ DX
and
2)
X
p(x) = 1.
x∈DX
A probability distribution can be expressed in a table, or with the help of a formula, as
the examples below illustrate.
Example:
Let
X : number of children in a random family of a given village.
Assume that it has the probability distribution
x
0
1
2
3
4 Total
p(x) 0.1 0.3 0.4 0.15 0.05
1
A probability distribution can be illustrated by a histogram:
p(x)
6
0.4
0.3
0.2
0.1
0
1
2
3
x
4
Example: Consider an urn with 9 green balls and 7 yellow balls.
Assume that 4 balls are picked without replacement. Let
X : number of green balls picked.
Then
9
7
x 4−x
p(x) =
16
4
for x = 0, 1, 2, 3, 4.
Example: Same as above, except with replacement.
Then
x 4−x
4
9
7
p(x) =
x
16
16
for x = 0, 1, 2, 3, 4.
The probability distributions of the last two examples are illustrated below.
p(x)
p(x)
6
6
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
1
2
3
Without Replacement
4
x
0
1
2
3
With Replacement
4
x