OCE301 Part VI: Data Analysis

1
OCE301 Part VI Data Analysis Probability
Theorylecture 1
2
Reading Assignment

• Textbook Kreyszig, 8th edition,
• pp. 1050-1069.

3
Stem-and-Leaf Plot
85 86 82 84 88 87 85 86
88 81 86 81
n 12
3 groups 81-83 84-86 87-89
81 81 82 84 85 85 86 86
86 87 88 88
cumulative relative frequencies
4
Histogram
h80round(5(randn(1,100)))
hist(h,10)
number of occurrences
absolute frequency
h
5
Histogram Relative Frequency
bar(x,n/sum(n))
n,xhist(h,10)
relative frequency
h
6
Matlab hist
N hist(Y) bins the elements of Y into 10
equally spaced containers and returns the
number of elements in each container.
N hist(Y,M), where M is a scalar, uses M bins.
N hist(Y,X), where X is a vector, returns the
distribution of Y among bins with centers
specified by X.
(so called class marks)
7
Histogram Using Class Marks
nhist(h,x)
bar(x,n)
x68492
8
Mean, Variance, Standard Deviation
mean
variance
Matlab functions mean, var, std
9
Experiments, Outcomes, Events
Experiment a process of measurement or
observation
Trial a single performance of an experiment
Outcome (sample point) the result from a trial
Sample space (S) the set of all possible outcomes
(outcomes simple events)
Events the subset of S
Examples
Rolling a dice. S 1,2,3,4,5,6 Events
A1,3,5 B5,6 etc. Simple events are
1,2,3,4,5,6
10
Unions and Intersections of Events
the union of A and B consists of all points that
are in A or B
Venn diagram
the intersection of A and B consists of all
points that are in A and B
11
Unions and Intersections example
Example
Event A 1, 2, 3
Event B 2, 4, 6
1, 2, 3, 4, 6
2
12
Mutually Exclusive (or Disjoint)
If A and B have no points in common
A and B are mutually exclusive (or disjoint)
Example
Event A 1, 3, 5
Event B 2, 4, 6
13
Complements of Events
complement of A consists of all the points of S
that are not in A
Ac
14
De Morgans Law
15
Probability
P(A) probability of an event A
Example
In rolling a fair die, what is the probability
of A being an even number.
Sample space S 1, 2, 3, 4, 5, 6
Event A 2, 4, 6
16
Axioms of Probability
17
Basic Theorems
Complementation rule
Addition rule for arbitrary events
18
Conditional probability
The probability of an event B under the
condition that an event A occurs
19
Independent Events
Events A and B are called independent events, if
20
Example Problem 22.3 (prob.3)
If a box contains 10 left-handed and 20
right-handed screws, what is the probability of
obtaining at least one right-handed screw in
drawing 2 screws with replacement?
probability that both are left-handed
probability that at least one is right-handed
21
Example Problem 22.3 (prob.5)
Three screws are drawn at random from a lot of
100 screws, 10 of which are defective. Find the
probability of the event that all 3 screws
drawn are non-defective, assuming that we drawn
(a) with replacement, (b) without replacement
(apply conditional probability)
22
Permutations
Permutation an ordered arrangement of a set of
objects
Three letters a,b,c
6 possible permutations abc, acb, bac, bca, cab,
cba
3! 321 6
The number of permutations of n different things
taken all at a time is
n! n(n-1)21
Read n factorial
23
Permutations (contd.)
Three letters a,a,c
3 possible permutations aac, aca, caa
If n given things can be divided into c classes
of alike things differing from class to class,
then the number of permutations of these things
taken all at a time is
n1 n2 nc n
n1 3
3! permutations of aaa are viewed to be one
24
Permutations (contd.)
The number of different permutations of n
different things taken k at a time without
repetitions is
1 2 3
k
n n-1 n-2
n-k1
with repetitions
1 2 3
k
n n n
n
25
Combinations
Permutation order is essential
Combination order is not essential
a,b,c and b,c,a are different permutation,
but they are the same combination.
Example
The number of different combinations of n
different things taken k at a time without
repetitions is
(k! permutations are corresponding to 1
combination)
26
Combinations (contd.)
The number of different combinations of n
different things taken k at a time with
repetitions is
27
Extra Credit 10 points
Proved by induction
The number of different combinations of n
different things taken k at a time with
repetitions is
28
Random Variables
The quantity that we observe in an experiment is
called a random variable ( or stochastic
variable) because the value it will assume in
the next trial depends on chance, on randomness.
Example
If we roll a dice, we get one of the numbers
from 1 to 6, but we dont know which one will
show up next.
29
Definition Random Variable
Use probability function to characterize a random
variable
30
Rolling a Fair Die
Probability mass function
discrete random variable
Total probability is equal to 1
31
Rotating an Unbiased Disk
angle q to which the indicator points
Probability density function
continuous random variable
Total probability is equal to 1