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Random Variables and
Stochastic Processes – 0903720
Dr. Ghazi Al Sukkar
Email: ghazi.alsukkar@ju.edu.jo
Office Hours: will be posted soon
Course Website:
http://www2.ju.edu.jo/sites/academic/ghazi.alsukkar
Most of the material in these slide are based on slides prepared by Dr. Huseyin Bilgekul
http://faraday.ee.emu.edu.tr/ee571/
1
Cumulative Distribution Function (CDF)
•
Since the elements of S that are contained in the event
𝑋 ≤ 𝑥 changes as the number 𝑥 takes various values,
then 𝑃 𝑋 ≤ 𝑥 is a number that depends on 𝑥.
• Definition
If X is a random variable define over S, then the
function F defined on (, ) by
𝐹𝑋 𝑥 = 𝑃 𝜁 ∈ 𝑆: 𝑋(𝜁) ≤ 𝑥 = 𝑃(𝑋  𝑥)
is called the distribution function or cumulative
distribution function (CDF) of X.
• The CDF gives a complete description of the
random variable.
2
• Example: In the die experiment, the random variable is
such that 𝑋 𝜁𝑖 = 𝑖, if the die is fair, then the CDF is a
staircase function.
• 𝐹 9 =𝑃 𝑋≤9 =𝑃 𝑆 =1
• 𝐹 3 = 𝑃 𝑋 ≤ 3 = 𝑃 𝜁1 , 𝜁2 , 𝜁3 =
3
6
• 𝐹 3 = 𝑃 𝑋 ≤ 3.4 = 𝑃 𝜁1 , 𝜁2 , 𝜁3 =
3
6
3
Properties of CDF
Properties
1. 0 ≤ 𝐹𝑋 (𝑥) ≤ 1 (this follows from Axiom 1 of probability).
2. 𝐹𝑋 (𝑥) is nondecreasing, 𝐹𝑋 𝑥1 ≤ 𝐹𝑋 𝑥2 if 𝑥1 < 𝑥2 (this is
because event 𝑋 ≤ 𝑥1 ⊂ 𝑋 ≤ 𝑥2 ).
3. 𝐹𝑋 +∞ = 1, since 𝑋 ≤ ∞ = 𝑆.
4. 𝐹𝑋 −∞ = 0, since 𝑋 ≤ −∞ = ∅.
5. If 𝐹𝑋 𝑥𝑜 = 0 then, 𝐹𝑋 𝑥𝑜 = 0, ∀𝑥 ≤ 𝑥𝑜 .
6. 𝑃 𝑋 > 𝑥 = 1 − 𝐹𝑋 (𝑥), since 𝑋 ≤ 𝑥 ∪ 𝑋 > 𝑥 = 𝑆 and
they are mutually exclusive events.
7. 𝐹𝑋 (𝑥) is continuous from the right. lim 𝐹𝑋 (𝑥 +
𝜖→0
𝜖>0
4
8. 𝑃 𝑥1 < 𝑋 ≤ 𝑥2 = 𝐹𝑋 𝑥2 − 𝐹𝑋 𝑥1 . since 𝑋 ≤
5
Discrete-Type RV
• 𝑋 is a discrete RV if 𝐹𝑋 (𝑥) is a staircase
function ⟹ 𝑥𝑖 is a discontinuity point.
−
• 𝑃 𝑋 = 𝑥𝑖 = 𝐹𝑋 𝑥𝑖 − 𝐹𝑋 𝑥𝑖 = 𝑝𝑖
• 𝐹𝑋 𝑥 = 𝑖𝑘=−∞ 𝑝𝑘 , 𝑥𝑖 ≤ 𝑥 < 𝑥𝑖+1
• 𝑘 𝑝𝑘 = 1
𝑝𝑖
𝑥𝑖
6
Continuous-Type RV
• 𝑋 is a continuous RV if 𝐹𝑋 (𝑥) is a continuous
function.
−
• 𝐹𝑋 𝑥 = 𝐹𝑋 𝑥 ⟹ 𝑃 𝑋 = 𝑥 = 0
7
• Mixed RV.
8
Percentiles
• The 𝑢 percentile of an RV 𝑋 is the smallest
number 𝑥𝑢 such that:
𝑢 = 𝑃 𝑋 ≤ 𝑥𝑢 = 𝐹𝑋 (𝑥𝑢 )
• 𝑥𝑢 is the inverse of the function 𝑢 = 𝐹𝑋 (𝑥)
• The median of 𝑋 is the smallest number 𝑚
such that 𝐹𝑋 𝑚 = 0.5 ⟹ 𝑚 𝑖𝑠 𝑡ℎ𝑒 50%
percentile of 𝑋.
9
Example
The distribution function of a random variable X is given by
0
x
4


F ( x)   12
1 x 1
2
 12

1
x0
0  x 1
1 x  2
2 x3
x3
Compute the following quantities：
(a) P(X < 2)
(b) P(X = 2)
(d) P(X > 3/2)
(e) P(X = 5/2)
(c) P(1 X < 3)
(f) P(2<X  7)
10
Example
For the experiment of flipping a fair coin twice, let
X be the number of tails and calculate F(t), the
distribution function of X, and then sketch its
graph.
Sol：
0
1 / 4

Ans : F (t )  
3 / 4
1
t  0,
0  t  1,
1  t  2,
t  2.
11
Example
Suppose that a bus arrives at a station every day
between 10:00 Am and 10:30 AM, at random. Let X
be the arrival time; find the distribution function
of X, F(t), and then sketch its graph.
Sol：
t  10,
0

Ans : F (t )  2(t  10) 10  t  10.5,
1
t  10.5.

12
Example
The sales of a convenience store on a randomly selected
day are X thousand dollars, where X is a random variable
with a distribution function of the following form：
t 0
0
1 t 2
0  t 1
2
F (t )  
2
k
(
4
t

t
) 1 t  2


t  2.
1
Suppose that this convenience store’s total sales on any
given day are less than $2000.
(a) Find the value of k.
(b) Let A and B be the events that tomorrow the store’s
total sales are between 500 and 1500 dollars, and
over 1000 dollars, respectively. Find P(A) and P(B).
(c) Are A and B independent events?
13
Probability Density Function (pdf)
• The pdf is defined as the derivative of the cdf:
𝑑𝐹𝑋 (𝑥)
𝑓𝑋 (𝑥) ≜
𝑑𝑥
𝑥
⟹ 𝐹𝑋 𝑥 =
Since
𝑑𝐹𝑋 (𝑥)
𝑑𝑥
=
𝑓𝑋 𝑢 𝑑𝑢
−∞
𝐹𝑋 𝑥+∆𝑥 −𝐹𝑋 (𝑥)
lim
∆𝑥
𝑥→0
≥ 0,
from the monotone-nondecreasing nature of
𝐹𝑋 (𝑥), it follows that 𝑓𝑋 (𝑥) ≥ 0 for all 𝑥.
14
• It follows that:
𝑥2
𝑃 𝑥1 < 𝑋 ≤ 𝑥2 = 𝐹𝑋 𝑥2 − 𝐹𝑋 𝑥1 =
𝑓𝑋 𝑥 𝑑𝑥
𝑥1
FX (x)
1
• 𝑃 𝑥 ≤ 𝑋 ≤ 𝑥 + ∆𝑥 ≅ 𝑓𝑋 (𝑥)∆𝑥
Provided that ∆𝑥 is sufficiently small.
x
x1 x2
f X (x)
x
x1 x2
• The probability that 𝑋 is in a small interval ∆𝑥 is proportional to
𝑓𝑋 (𝑥) and it is maximum if that interval contains the point 𝑥𝑚 ,
where𝑓𝑋 (𝑥) is maximum.
𝑥𝑚 is called the mode (most likely value of 𝑋).
Note: an RV is called unimodal if it has a single mode.
15
• Basic properties of pdf:
1. 𝑓𝑋 (𝑥) ≥ 0.
2.
∞
𝑓
−∞ 𝑋
𝑥 𝑑𝑥 = 1.
3. In general, 𝑃 𝑋 ∈ 𝐴 =
𝐴
𝑓𝑋 𝑥 𝑑𝑥.
16
Pdf for Discrete Random Variables
The pfd for discrete RV is:
𝑓𝑋 𝑥 = 𝑖 𝑝𝑖 𝛿(𝑥 − 𝑥𝑖 )
It is known as probability mass function (pmf).
𝑝𝑖 = 𝑃 𝑋 = 𝑥𝑖
𝑥𝑖 s represent the discontinuity points.
Again: 𝐹𝑋 𝑥 = 𝑖𝑘=−∞ 𝑝𝑘 , 𝑥𝑖 ≤ 𝑥 < 𝑥𝑖+1
𝑘 𝑝𝑘 = 1
f X (x )
pi
xi
x
17
Example
In the experiment of rolling a balanced die twice,
let X be the maximum of the two numbers
obtained. Determine and sketch the probability
mass function and the distribution function of X.
Sol：
x  1,
0
1 / 36

4 / 36

Ans : F ( x)  9 / 36
16 / 36

25 / 36

1
1  x  2,
2 x3
3  x  4.
4  x  5,
5  x  6,
x  6.
18
Example
Can a function of the form
c( 23 ) x x  1,2,3,...
p ( x)  
elsewhere.
 0
be a probability mass function ?
Sol：
19
Example
Let X be the number of births in a hospital until
the first girl born. Determine the probability
mass function and the distribution function of X.
Assume that the probability is 1/2 that a baby
born is a girl.
Sol：
t  1,
0
Ans : F (t )  
n 1
1

(
1
/
2
)
n  1  t  n, n  2,3,4,.

20