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RAJEEV GANDHI MEMORIAL COLLEGE OF
ENGINEERING & TECHNOLOGY
(AUTONOMOUS)
Subject
Course
Name of the Faculty
: Probability and Statistics
: II B.Tech
: 1) Prof S. Venkateswarlu
2) P. Sreedevi
Unit 1: Basic Probability
Outcomes:
Explain basic probability concepts and definitions
Apply common rules of probability
Compute conditional probabilities
Determine whether events are statistically independent
Definitions:
•Probability – The chance or likelihood that an uncertain
(particular) event will occur
•Probability is always between 0 and 1, inclusive
•There are three approaches to assessing the probability
of un uncertain event:
1. A priori classical probability-based of a prior knowledge
prob. of occurrence 
X
number of occurance of the event

n
total number of possible outcomes
2. Empirical classical probability- based on observed data
prob. of occurrence 
number of favorable outcomes observed
total number of outcomes observed
3. Subjective probability - an individual judgment or opinion
about the probability of occurrence
Basic Concepts:
•
Sample Space – the collection of all possible events
•
An Event – Each possible type of occurrence or
outcome from the sample space
•
Simple Event – an event that can be described by a
single characteristic
•
Complement of an event A -- All outcomes that are not
part of event A
Joint event --Involves events that can be described by
two or more characteristics simultaneously
•
Rules of Probability:
1.
General Addition Rule
P(A or B) = P(A) + P(B) - P(A and B)
If A and B are mutually exclusive, then
P(A and B) = 0, so the rule can be simplified
P(A or B) = P(A) + P(B)
If A and B are collectively exhaustive, then
P(A or B) = P(A) + P(B)=1
2. A conditional probability is the probability of
one event, given that another event has
occurred:
P(A and B)
P(A | B) 
P(B)
P(A and B)
P(B | A) 
P(A)
Where P(A and B) = joint probability of A and B
P(A) = marginal probability of A
P(B) = marginal probability of B
The conditional
probability of A
given that B has
occurred
The conditional
probability of B
given that A has
occurred
3.
Multiplication Rule– for two events A and B
P(A and B)  P(A | B) P(B)
3.
Two events A and B are statistically independent if the
probability of one event is unchanged by the
knowledge that other even occurred. That is:
P(A | B)  P(A) or P(B | A)  P(B)
4.
Then the multiplication rule for two statistically
independent events is:
P(A and B)  P(A) P(B)
Random Variables
• Random Variable (RV): A numeric outcome that results
from an experiment
• For each element of an experiment’s sample space, the
random variable can take on exactly one value
• Discrete Random Variable: An RV that can take on only a finite
or countably infinite set of outcomes
• Continuous Random Variable: An RV that can take on any value
along a continuum (but may be reported “discretely”)
• Random Variables are denoted by upper case letters (Y)
Individual outcomes for an RV are denoted by lower case
letters (y)
Probability (Mass) Function :
p ( y )  P (Y  y )
p ( y )  0 y
 p( y)  1
all y
Cumulative Distribution Function (CDF) :
F ( y )  P (Y  y )
F (b)  P (Y  b) 
b
 p( y)
y  
F ( )  0 F ()  1
F ( y ) is monotonically increasing in y
Example – Rolling 2 Dice (Red/Green)
Y = Sum of the up faces of the two die. Table gives value of y for all elements in S
Red\Green
1
2
3
4
5
6
1
2
3
4
5
6
7
2
3
4
5
6
7
8
3
4
5
6
7
8
9
4
5
6
7
8
9
10
5
6
7
8
9
10
11
6
7
8
9
10
11
12
Random Variables
• Random Variable (RV): A numeric outcome that results from
an experiment
• For each element of an experiment’s sample space, the random
variable can take on exactly one value
• Discrete Random Variable: An RV that can take on only a
finite or countably infinite set of outcomes
• Continuous Random Variable: An RV that can take on any
value along a continuum (but may be reported “discretely”)
• Random Variables are denoted by upper case letters (Y)
• Individual outcomes for an RV are denoted by lower case
letters (y)
Probability Distributions
• Probability Distribution: Table, Graph, or Formula that describes
values a random variable can take on, and its corresponding
probability (discrete RV) or density (continuous RV)
• Discrete Probability Distribution: Assigns probabilities (masses)
to the individual outcomes
• Continuous Probability Distribution: Assigns density at individual
points, probability of ranges can be obtained by integrating
density function
• Discrete Probabilities denoted by: p(y) = P(Y=y)
• Continuous Densities denoted by: f(y)
• Cumulative Distribution Function: F(y) = P(Y≤y)
Rolling 2 Dice – Probability Mass Function & CDF
y
p(y)
F(y)
2
1/36
1/36
3
2/36
3/36
4
3/36
6/36
5
4/36
10/36
6
5/36
15/36
7
6/36
21/36
8
5/36
26/36
9
4/36
30/36
10
3/36
33/36
11
2/36
35/36
12
1/36
36/36
# of ways 2 die can sum to y
p( y) 
# of ways 2 die can result in
y
F ( y )   p (t )
t 2
Rolling 2 Dice – Probability Mass Function
Dice Rolling Probability Function
0.18
0.16
0.14
0.12
p(y)
0.1
0.08
0.06
0.04
0.02
0
2
3
4
5
6
7
y
8
9
10
11
12
Rolling 2 Dice – Cumulative Distribution Function
Dice Rolling - CDF
1
0.9
0.8
0.7
F(y)
0.6
0.5
0.4
0.3
0.2
0.1
0
1
2
3
4
5
6
7
y
8
9
10
11
12
13
Expected Values of Discrete RV’s
• Mean (aka Expected Value) – Long-Run average value
an RV (or function of RV) will take on
• Variance – Average squared deviation between a
realization of an RV (or function of RV) and its mean
• Standard Deviation – Positive Square Root of Variance
(in same units as the data)
• Notation:
– Mean: E(Y) = m
– Variance: V(Y) = s2
– Standard Deviation: s
Expected Values of Discrete RV’s
Mean : E (Y )  m   yp ( y )
all y
Mean of a function g (Y ) : E g (Y )   g ( y ) p ( y )
all y

 

p ( y )    y  2 ym  m  p ( y ) 
Variance : V (Y )  s 2  E (Y  E (Y )) 2  E (Y  m ) 2 
  ( y  m )2
all y
2
2
all y
  y p ( y )  2 m  yp ( y )  m
2
all y
 
all y
2
 p( y ) 
all y
 
 E Y 2  2 m ( m )  m 2 (1)  E Y 2  m 2
Standard Deviation : s   s 2
Expected Values of Linear Functions of Discrete RV’s
Linear Functions : g (Y )  aY  b (a, b  constants )
E[aY  b]   (ay  b) p ( y ) 
all y
 a  yp ( y )  b p ( y )  am  b
all y
all y
V [aY  b]   (ay  b)  (am  b)  p ( y ) 
2
all y
 ay  am 
2
all y
a
2
 ( y  m)
all y
s aY b  a s


p( y )   a  y  m  p( y ) 
2
2
all y
2
p( y)  a s
2
2
Example – Rolling 2 Dice
y
p(y)
yp(y)
y2p(y)
2
1/36
2/36
4/36
3
2/36
6/36
18/36
4
3/36
12/36
48/36
5
4/36
20/36
100/36
6
5/36
30/36
180/36
7
6/36
42/36
294/36
8
5/36
40/36
320/36
9
4/36
36/36
324/36
10
3/36
30/36
300/36
11
2/36
22/36
242/36
12
1/36
12/36
144/36
Sum 36/36
=1.00
252/36
=7.00
1974/36=
54.833
12
m  E (Y )   yp( y )  7.0
y 2
s 2  E Y 2  m 2   y 2 p( y )  m 2
12
y 2
 54.8333  (7.0) 2  5.8333
s  5.8333  2.4152
Discrete Uniform Distribution
• Suppose Y can take on any integer value between a and b inclusive,
each equally likely (e.g. rolling a dice, where a=1 and b=6). Then Y
follows the discrete uniform distribution.
f ( y) 
1
b  (a  1)
a yb
0
ya
 int ( y )  (a  1)
F ( y)  
a  y  b int( x)  integer portion of x
b

(
a

1
)

1
yb
b
a 1 
b


1
1
1
 b(b  1) (a  1)a  b(b  1)  a (a  1)
 
E (Y )   y
y

y 


 2  2   2(b  (a  1))
b

(
a

1
)
b

(
a

1
)
b

(
a

1
)
y a 
y 1 

 y 1
b
 b 2 a 1 2 


1
1
1
 b(b  1)( 2b  1) (a  1)a (2a  1) 
 
E Y 2   y 2 

 y   y  


b

(
a

1
)
b

(
a

1
)
b

(
a

1
)
6
6


y a
y 1


 y 1

 

b(b  1)( 2b  1)  a (a  1)( 2a  1)
6(b  (a  1))
b(b  1)( 2b  1)  a (a  1)( 2a  1)  b(b  1)  a (a  1) 
 V (Y )  E Y  E (Y ) 


6(b  (a  1))
 2(b  (a  1)) 
 
2
2
Note : When a  1 and b  n :
E (Y ) 
n 1
2
V (Y ) 
(n  1)( n  1)
12
s
(n  1)( n  1)
12
2