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Lecture 3 B
Maysaa ELmahi
3.3. Distribution Functions of Continuous Random Variables
Recall that a random variable X is said to be continuous if its space is
either an interval or a union of intervals.
Definition 3.7.
Let X be a continuous random variable whose space is the set of real numbers I R.
A nonnegative real valued function f : IR
IR is said to be the probability
density function for the continuous random variable X if it satisfies:
(a)
∞
𝐟
−∞
𝐱 𝐝𝐱 = 𝟏
, and
(b) if A is an event, then 𝐩 𝐀 =
𝐀
𝐟 𝐱 𝐝𝐱
Example 3.10.
Is the real valued function f : IR
𝟐𝐱 −𝟐
IR defined by
𝐢𝐟 𝟏 < 𝐱 < 𝟐
𝐟 𝐱 =
𝟎
𝐨𝐭𝐡𝐞𝐫𝐰𝐢𝐬𝐞
(a) probability density function for some random variable X?
Answer:
∞
𝟐
𝟐𝐱 −𝟐 𝐝𝐱
𝐟 𝐱 𝐝𝐱 =
−∞
𝟏
= −𝟐
𝟏 𝟐
𝐱 𝟏
−𝟐
𝟏
𝟐
− 𝟏 =1
Thus f is a probability density function.
Example 3.11.
Is the real valued function f : IR
𝟏+ 𝐱
IR defined by
𝐢𝐟 − 𝟏 < 𝐱 < 𝟏
𝐟 𝐱 =
𝟎
𝐨𝐭𝐡𝐞𝐫𝐰𝐢𝐬𝐞
(a) probability density function for some random variable X?
Answer:
∞
1
𝑓 𝑥 𝑑𝑥 =
(1 + 𝑥 ) 𝑑𝑥
−∞
−1
0
=
1
1 − 𝑥 𝑑𝑥 +
−1
0
1
= 𝑥 − 2 𝑥2
=1 +
(1 + 𝑥) 𝑑𝑥
1
2
0
+
−1
1
1
𝑥 + 2 𝑥2
+1+2 = 3
1
0
Definition 3.8.
Let f(x) be the probability density function of a continuous random variable X.
The cumulative distribution function F(x) of X is defined as
𝐅 𝐱 =𝐏 𝐗≤𝐱
=
𝐱
𝐟
−∞
𝐭 𝐝𝐭
Theorem 3.5.
If F(x) is the cumulative distribution function of a continuous random variable X,
the probability density function f(x) of X is the derivative of F(x), that is
𝐝
𝐅 𝐱 = 𝐟(𝐱)
𝐝𝐱
Theorem 3.6.
Let X be a continuous random variable whose c d f is F(x). Then followings are
true:
𝑎 . 𝑃 𝑋 < 𝑥 = 𝐹(𝑥)
𝑏 . 𝑃 𝑋 > 𝑥 = 1 − 𝐹(𝑥)
𝑐 .𝑃 𝑋 = 𝑥 = 0
𝑑 .𝑃 𝑎 < 𝑋 < 𝑏 =
𝐹(𝑏) − 𝐹(𝑎)
Example : (H.W)
𝑘 + 1 𝑥2
𝑖𝑓
0<𝑥<1
𝑓 𝑥 =
0
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
a. what is the value of the constant k?
b. What is the probability of X between the first and third?
d. What is the cumulative distribution function?
4.2. Expected Value of Random Variables
Definition 4.2.
Let X be a random variable with space 𝑅𝑥 and probability density function f(x).
The mean 𝜇𝑥 of the random variable X is defined as
𝒙𝒇(𝒙)
𝒊𝒇 X is discrete
𝒙∈𝑹𝑿
𝛍𝐱 = 𝐄(𝐱) =
∞
𝒙 𝒇 𝒙 𝒅𝒙
−∞
if X is continuous
Example :
x
0
1
2
3
P(x)
1/8
3/8
3/8
1/8
what is the mean of X?
Answer:
  E (x )   x f (x )
x
= 0* 1/8+ 1* 3/8 +2* 3/8 +3 * 1/8
= 0 + 3/8 + 6/8 + 3/8 = 12/8
Example :
𝟏
𝟓
𝒇 𝒙 =
𝟎
𝒊𝒇
𝟐<𝒙<𝟕
𝒐𝒕𝒉𝒆𝒓𝒘𝒊𝒔𝒆
Answer:

  E (x )   x f (x )dx

7
=
2
1
𝑥 𝑑𝑥
5
=
1 2 7
𝑥 2
10
=
1
10
45
9
49 − 4 = 10 =2
Theorem 4.1
Let X be a random variable with p d f f(x). If a and b are any two real numbers, then
𝐚. 𝐄(𝐚𝐱 + 𝐛 ) = a 𝐄 𝐱 + 𝐛
b. 𝐄(𝐚𝐱) = 𝐚 𝐄(𝐱)
c. 𝐄(𝐚) = 𝐚
4.3. Variance of Random Variables
Definition 4.4.
Let X be a random variable with mean 𝜇𝑥 . The variance of X, denoted by
Var(X), is defined as
𝐕𝐚𝐫 𝐱 = ( 𝐄 𝐱 − 𝛍𝐱 )
𝟐
𝛔𝟐 𝐱 = 𝐄 𝐱 𝟐 − (𝛍𝟐 𝐱 )
Example :
2(𝑥 − 1)
𝑖𝑓
1<𝑥<2
𝑓 𝑥 =
0
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
a. what is the variance of X?
Answer:
∞
𝑥
−∞
𝜇𝑥 = 𝐸 𝑥 =
𝑓 𝑥 𝑑𝑥 =
=2
2 2
(𝑥 −𝑥)𝑑𝑥
1
=2
𝑥3
3
−
=2
8
3
−2
=2∗
2(𝑥 − 1)𝑑𝑥
𝑥2 2
2 1
4
5
6
2
𝑥
1
=
−
10
6
1
3
1
4
1
− 2 = 2((6 - ( - 6 ) )
∞
𝐸 𝑥
2
2
2
=
𝑥 2 2(𝑥 − 1)𝑑𝑥
𝑥 𝑓 𝑥 𝑑𝑥 =
−∞
1
= 2
2 3
2
(𝑥
−(𝑥
)𝑑𝑥
1
=2
𝑥4
4
−
=2
16
4
−3
𝑥3 2
3 1
8
17
−
1
4
1
16
1
− 3 = 2((12 - ( - 12 ) )
= 2((12) ) = 17/6
Thus, the variance of X is given by
𝛔𝟐 𝐱 = 𝐄 𝐱 𝟐 − (𝛍𝟐 𝐱 )
=
17
6
–
10
6
=
Remark:
Var (𝑎𝑥 + 𝑏 ) = 𝑎2 𝑉𝑎𝑟 𝑥
𝑉𝑎𝑟 (𝑎 𝑥 ) =𝑎2 𝑉𝑎𝑟 𝑥
𝑉𝑎𝑟 (𝑎) =0
7
6
4.1. Moments of Random Variables
Definition 4.1.
The nth moment about the origin of a random variable X, as denoted by E(𝑥 𝑛 ),
is defined to be
𝐱 𝐧 𝐟(𝐱)
𝐢𝐟 X is discrete
𝐱∈𝐑 𝐗
𝐄 𝐱𝐧
=
∞
𝐱 𝐧 𝐟 𝐱 𝐝𝐱
−∞
if X is continuous
for n = 0, 1, 2, 3, ...., provided the right side converges absolutely. If n = 1, then E(X)
is called the first moment about the origin.
If n = 2, then E(𝑥 2 ) is called the second moment of X about the origin.
4.5. Moment Generating Functions
Definition 4.5.
Let X be a random variable whose probability density function is f(x).
A real valued function M : IR
IR defined by
𝑴 𝒕 = 𝑬(𝒆𝒕𝒙 )
is called the moment generating function of X if this expected value exists
for all t in the interval −h < t < h for some h > 0.
Using the definition of expected value of a random variable, we obtain
the explicit representation for M(t) as
𝐞𝐭𝐱 𝐟(𝐱)
𝐢𝐟 X is discrete
𝐱∈𝐑 𝐗
𝐌(𝐭) =
∞
𝐞𝐭𝐱 𝐟 𝐱 𝐝𝐱
−∞
if X is continuous