1. Lecture 3 B
Maysaa ELmahi

2. 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

3. Example 3.10.
Is the real valued function f : IR IR defined by
𝐟 𝐱 = 𝟐𝐱 −𝟐 𝐢𝐟 𝟏<𝐱<𝟐 𝟎 𝐨𝐭𝐡𝐞𝐫𝐰𝐢𝐬𝐞
(a) probability density function for some random variable X?
Answer:
−∞ ∞ 𝐟 𝐱 𝐝𝐱= 𝟏 𝟐 𝟐𝐱 −𝟐 𝐝𝐱
=−𝟐 𝟏 𝐱 𝟐 𝟏

4. −𝟐 𝟏 𝟐 −𝟏 =1
Thus f is a probability density function.
Example 3.11.
Is the real valued function f : IR IR defined by
𝐟 𝐱 = 𝟏+ 𝐱 𝐢𝐟 −𝟏<𝐱<𝟏 𝟎 𝐨𝐭𝐡𝐞𝐫𝐰𝐢𝐬𝐞
(a) probability density function for some random variable X?

5. Answer:
−∞ ∞ 𝑓 𝑥 𝑑𝑥= −1 1 (1+ 𝑥 ) 𝑑𝑥
= −1 0 1−𝑥 𝑑𝑥+ 0 1 (1+𝑥) 𝑑𝑥
= 𝑥− 1 2 𝑥 − 𝑥 𝑥
= = 3

6. 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 𝐝𝐱 𝐅 𝐱 =𝐟(𝐱)

7. 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)

8. Example : (H.W)
𝑓 𝑥 = 𝑘+1 𝑥 𝑖𝑓 <𝑥< 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?

9. 4.2. Expected Value of Random Variables
Definition 4.2.
Let X be a random variable with space 𝑅 𝑥 and probability density function f(x).
The mean 𝜇 𝑥 of the random variable X is defined as
𝛍 𝐱 =𝐄(𝐱) = 𝒙∈ 𝑹 𝑿 𝒙𝒇(𝒙) 𝒊𝒇 X is discrete −∞ ∞ 𝒙 𝒇 𝒙 𝒅𝒙 if X is continuous

10. Example : x 1 2 3
P(x)
1/8
3/8
what is the mean of X?
Answer:
= 0* 1/8+ 1* 3/8 +2* 3/8 +3 * 1/8
= / /8 + 3/8 = 12/8

11. Example :
𝒇 𝒙 = 𝟏 𝟓 𝒊𝒇 𝟐<𝒙<𝟕 𝟎 𝒐𝒕𝒉𝒆𝒓𝒘𝒊𝒔𝒆
Answer:
= 2 7 𝑥 1 5 𝑑𝑥
= 𝑥
= −4 = = 9 2

12. 4.3. Variance of Random Variables
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 𝜇 𝑥 . The variance of X, denoted by Var(X), is defined as

13. 𝐕𝐚𝐫 𝐱 = ( 𝐄 𝐱 − 𝛍 𝐱 ) 𝟐
𝛔 𝟐 𝐱 =𝐄 𝐱 𝟐 − (𝛍 𝟐 𝐱 )
Example :
𝑓 𝑥 = 2(𝑥−1) 𝑖𝑓 <𝑥< 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
a. what is the variance of X?
Answer:

14. 𝜇 𝑥 =𝐸 𝑥 = −∞ ∞ 𝑥 𝑓 𝑥 𝑑𝑥= 1 2 𝑥 2(𝑥−1)𝑑𝑥
= (𝑥 2 −𝑥)𝑑𝑥
= 2 𝑥 − 𝑥
= − − − = 2(( ( ) )
=2∗ =

15. 𝐸 𝑥 2 = −∞ ∞ 𝑥 2 𝑓 𝑥 𝑑𝑥= 𝑥 2 2(𝑥−1)𝑑𝑥
= (𝑥 3 − (𝑥 2 )𝑑𝑥
= 2 𝑥 − 𝑥
= − − − = 2(( ( ) )
= 2(( ) ) = 17/6

16. Thus, the variance of X is given by
𝛔 𝟐 𝐱 =𝐄 𝐱 𝟐 − (𝛍 𝟐 𝐱 )
= – =
Remark:
Var (𝑎𝑥+𝑏 ) = 𝑎 2 𝑉𝑎𝑟 𝑥
𝑉𝑎𝑟 (𝑎 𝑥 ) = 𝑎 2 𝑉𝑎𝑟 𝑥
𝑉𝑎𝑟 (𝑎) =0

17. 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
𝐄 𝐱 𝐧 = 𝐱∈ 𝐑 𝐗 𝐱 𝐧 𝐟(𝐱) 𝐢𝐟 X is discrete −∞ ∞ 𝐱 𝐧 𝐟 𝐱 𝐝𝐱 if X is continuous

18. for n = 0, 1, 2, 3,. , provided the right side converges absolutely
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( 𝒆 𝒕𝒙 )
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.

19. Using the definition of expected value of a random variable, we obtain
the explicit representation for M(t) as
𝐌(𝐭) = 𝐱∈ 𝐑 𝐗 𝐞 𝐭𝐱 𝐟(𝐱) 𝐢𝐟 X is discrete −∞ ∞ 𝐞 𝐭𝐱 𝐟 𝐱 𝐝𝐱 if X is continuous