Download Chapter 2 Probability and Distrubition Functions of Random

Chapter 2
Probability and Distrubition Functions of Random Variables
1. Random Variables:
Definition1: The function
X : S → ℝ
w → X(w)
for a ∈ ℝ if {𝑤: 𝑋(𝑤) ≤ 𝑎} ∈ 𝜐
the function x is called a random variable.
w : the element of S
υ : the set of events
→ The set of X values, is represented by 𝐷𝑥
A random variable associates a red number with each element in the sample space .We can
go to the mathematical world from the real world by the random veriable and we can study in
the mathematical world easily.
Example: A coin is tossed in three times. In this experiment , x is defined as the number of
heads. Is the function x is a random variable and then, find these probabilities
P(𝑥 ≤ 1)=?
P(𝑥 > 3)=?
P(𝑥 ≤ 0.75)=?
First we obtain the sample space S
S = {𝑇𝑇𝑇, 𝑇𝑇𝐻, 𝐻𝑇𝑇, 𝑇𝐻𝑇, 𝑇𝐻𝐻, 𝐻𝑇𝐻, 𝐻𝐻𝑇, 𝐻𝐻𝐻}
𝑤1 𝑤2 𝑤3 𝑤4
𝑤5
𝑤6
𝑤7
𝑤8
x
ℝ
0
1
2
Now, for all a ∈ ℝ ,
a< 0 { 𝑤 ∶ 𝑋(𝑤) ≤ 𝑎} = ∅ ∈ 𝜐
3
0≤ 𝑎 < 1 {𝑤 ∶ 𝑋(𝑤) ≤ 𝑎} = {𝑤1 } ∈ 𝜐
1≤ 𝑎 < 2 {𝑤 ∶ 𝑋(𝑤) ≤ 𝑎} = {𝑤2, 𝑤3 , 𝑤4 } ∈ 𝜐
2≤ 𝑎 < 3 {𝑤 ∶ 𝑋(𝑤) ≤ 𝑎} = {𝑤5 , 𝑤6 , 𝑤7 } ∈ 𝜐
a≥ 3 {𝑤 ∶ 𝑋(𝑤) ≤ 𝑎} = S ∈ 𝜐
→ So we can say that the x is a random veriable easily.
Let’s find the probabilities,
P(𝑥 ≤ 1) = P(𝑥 = 0)+ P(𝑥 = 1)
= P({𝑤1 }) + P({𝑤2 , 𝑤3 , 𝑤4 })
=P ({𝑇𝑇𝑇}) + P({𝑇𝑇𝐻, 𝐻𝑇𝑇, 𝑇𝐻𝑇})
=
1
8
3
+
8
=
4
8
P(𝑥 > 3) = P (∅) = 0
1
P(𝑥 ≤ 0.75) = P(𝑥 = 0) = P({𝑤1 }) = 8
2. Probability Function and Distribution Function
Definition2:
X : S → ℝ
w → X(w)
X is a random variable and the set of X values 𝐷𝑥 , 𝐷𝑥 has a finite on cauntably infinite
number of points , the X is called the discrete random variable.
Definition3: ( Probability Function )
X is a discrete random variable
the function F(x)
f(𝑥) = P(𝑋 = 𝑥) ,
𝑥 ∈ 𝐷𝑥
is called the probability function of X
Example: A dice is tossed , random variable X is the number of points,
𝐷𝑥 = {1,2,3,4,5,6}
X is a discrete random variable,
the probability function of X
1
𝑥 ∈ 𝐷𝑥 = {1,2,3,4,5,6}
f(𝑥) = P(𝑋 = 𝑥) = 6
the graphic of function f
f(𝑥)
variable
x : the value of random
1
6
1
2
3
4
5
6
x
Definition4: The distribution function is represented by F(𝑥)
∀𝑥 ∈ ℝ
F :
ℝ → [0,1]
𝑥 → F(𝑥) = P(𝑋 ≤ 𝑥)
Example: A coin is tossed twice.The random variable X is , the number of to its.Let’s obtain
the probability function and the cumulative distribution function.
Solution: S = {𝐻𝐻, 𝐻𝑇, 𝑇𝐻, 𝑇𝑇}
x
0
ℝ
1
2
𝐷𝑥 = {0,1,2}
𝑓𝑥 (𝑥) = 𝑃(𝑋 = 𝑥)
𝑋=𝑥
0
P(𝑋 = 𝑥)
1
2
1
2
1
4
4
4
f(𝑥)
1
2
1
4
0
1
2
F : ℝ → [0,1]
𝑥 → 𝐹(𝑥) = 𝑃(𝑋 ≤ 𝑥)
𝑥<0
0≤𝑥<1
∀𝑥 ∈ ℝ
F(𝑥) = 𝑃( 𝑋 ≤ 𝑥)
= 𝑃(∅) = 0
F(𝑥) = 𝑃(𝑋 ≤ 𝑥)
= 𝑃(𝑥 = 0)
= 𝑃({𝐻𝐻}) =
1
4
1≤𝑥<2
F(𝑥) = 𝑃(𝑋 ≤ 𝑥)
= 𝑃(𝑥 = 0) + 𝑃(𝑥 = 1)
1 2
3
= + =
4 4
4
𝑥≥2
𝐹(𝑥) = 𝑃(𝑋 = 𝑥)
= 𝑃(𝑥 = 0) + 𝑃(𝑥 = 1) + 𝑃(𝑥 = 2) = 1
𝐹(𝑥)

1
3

4
1
4



1
2
Note
Examples 1. 2.
1. A dice is tossed in three times.
X: the number of heads. Obtain the probability function and the cumulative distribution
function.
Solution:
S = (𝑇𝑇𝑇 , 𝑇𝑇𝐻 , 𝑇𝐻𝑇 , 𝐻𝑇𝑇 , 𝑇𝐻𝐻 , 𝐻𝑇𝐻 , 𝐻𝐻𝑇 , 𝐻𝐻𝐻)
𝑤1 𝑤2
𝑤8
𝑋=𝑥
𝑓(𝑥)
𝑃(𝑋 = 𝑥)
0
1
2
3
1
3
3
1
8
8
8
8
1
𝑓(𝑥) = 𝑃(𝑋 = 𝑥) =
𝑥=0
8
3
𝑥=1
8
3
𝑥=2
8
1
8
𝑥<0
0
1⁄
8
0≤𝑥<1
𝐹𝑋 (𝑥) = 4⁄8
1≤𝑥<2
7⁄
8
2≤𝑥<3
1
𝑥≥3
𝑥=3