Transcript Document

WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam
WFM-6204: Hydrologic Statistics
Lecture-2: Probability and statistics
Akm Saiful Islam
Institute of Water and Flood Management (IWFM)
Bangladesh University of Engineering and Technology (BUET)
December, 2006
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam
Probability

If a random event can occur in n equal likely and
mutually exclusive ways, and if na of these ways
have an attribute A, then probability of the
occurrence of the event having attribute A is,
Prob(A) = na /n
 This

is known as priori definition
Probability can range between 0 and 1
 0 means the event never happens
 1 means it will always happen
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam
Example-1:


Find the probability of peak flow in excess of
1,00,000 cfs will occur.
Year
1907
1917
1927
Flow
66,300
111,000 93,700
1937
1957
112,000 88,700
1967
115,00
What should be the probability of this excess in 2
successive years ?
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam
Laws of probability

1. General Addition rule:
 If
A and B are two mutually inclusive events in S so
that A  B is not empty, then the probability of A or
B is given by:
p( A  B)  p( A)  p( B)  p( A  B)
notation  represents a union so that A  B
represents all elements in A or B or both. The
notation  represents an intersection so that A  B
represents all elements in both A and B.
 The
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam
Venn diagram
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam
 If
and are mutually exclusive events, then
both cannot occur and p.( A  B)  0 In this case,
p( A  B)  p( A)  p( B)
 If Ac
in A
represents all elements in S that are not
, then
A A  S
c
p( A  Ac )  p( A)  p( Ac )  1
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam

2. Multiplication rule:
 If
the probability of an event B depends on
the occurrence of an event A, then we write
pB A and say the probability of B given that A
has occurred. Thus the conditional probability
(the ratio) is given by:
p A  B 
pB A 
p A
 From
, where p ( A)  0
this equation, we find that
p( AB)  p( A) p( B / A)
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam

Occasionally, we must deal with more than two
events. If A, B and C are three non-mutually
exclusive events, then
p( ABC)  p( A) pB A pC AB
where p AB  0

The two events A and B are independent if
p( A  B)  p( A) p( B)
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam
Example-2:

A study of daily rainfall at Ashland, Kentucky,
has shown that in July the probability of a rainy
day is 0.444, a dry day following a dry day
0.724, a rainy day following a dry day is 0.276
and a dry day following a rainy day is 0.556. If it
is observed that a certain July day is rainy, what
is the probability that the next two days will also
be rainy?
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam
Probability theorems

1. Total probability theorem
 If B1 , B2 , .........,Bn
represents a set of mutually
exclusive and collectively exhaustive events,
one can determine the probability of another
event A from
n
p A   p A Bi  pBi 
i 1
Venn diagram for
total probability theorem
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam
Example-3:Total Probability theorem

It is known that the probability that the solar
radiation intensity will reach a threshold value of
0.25 for rainy days and 0.8 for non-rainy days. It
is also known that for this particular location the
probability of a rainy day is 0.36. What is the
probability the threshold intensity of solar
radiation will be reached?
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam


2. Bayes theorem
Bayes theorem deals with conditional probability.
It can be used to find p A B when the available
information is not compatible with that required to
apply the definition of conditional probability
directly. To accomplish this, the conditional
probability equation is rewritten as:
pB j A 
pB j pA B j 
p A

pB j  pA B j 
n
 p  A B  p B 
i 1
 Bayes
i
i
theorem provides a means of estimating
probabilities of one event by observing a second
event.
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam
Assignment-1:Baye’s theorem

The manager of a recreational facility has determined that
the probability he will have 1000 or more visitors on any
Sunday in July depends upon the maximum temperature
for that Sunday as shown in the following table. The table
also gives the probabilities that the maximum temperature
will fall in the indicated ranges. On a certain Sunday in
July, the facility has more than 1000 visitors. What is the
probability that the maximum temperature was in the
various temperature classes?
Temp (0F)
class
<60
60-70
70-80
80-90
90-100
>100
Prob of 1000 or more visitors
Prob of being in temp
0.05
0.20
0.50
0.75
0.50
0.25
0.05
0.15
0.20
0.35
0.20
0.05
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam
Random variables

A random variable is a function that associates
a real number with each element in the sample
space.

We use a capital letter, say X, to denote a
random variable and a corresponding small
letter, x , for one of its values. Any function of a
random variable is also a random variable. If X is
a random variable, then Z  f X  is also a random
variable.
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam




discrete random variable
 If the set of values a random variable can assume is
countable, the random variable is called a discrete
random variable.
continuous random variable.
 If the set of values a random variable can assume is
infinite, the random variable is said to be a
continuous random variable.
In most practical problems, continuous random variables
represent measured data, such as all possible
temperatures or the amount of rain received over a year,
whereas discrete random variables represent count
data, such as the number of rainy days experienced at a
particular location over a period of one year.
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam

The set of all possible outcomes of a random experiment is called the
sample space (S) of the experiment, because it usually consists of all the
things that can happen when one takes a sample. A sample space is often
defined based on the objective of the analysis.

A sample space is discrete if it consists of a finitely many or a countable infinite
set of outcomes. In the coin-tossing example, the sample space has two
outcomes and it is referred to as a finite sample space.
 If the elements/points of a sample space constitute a continuum - for example, all
the points on a line or all the points on a plane - the sample space is said to be a
continuous sample space. If a person is interested in the Nitrogen Oxide
emission of cars in grams per mile, the sample space would have to consist of all
the points on a continuous scale (a certain interval on the line of real numbers, of
which there is a continuum).


Each outcome in a sample space is called an element or a member of the
sample space or simply a sample point.
Thus, the sample space (S) of possible outcomes when a coin is tossed
may be written as:
S  H , T 


where, H and T corresponds to “heads” and “tails”, respectively.
Any subset A of a sample space is called an event. An event is a collection
of elements. The empty set  is called the impossible event; the subset S
is called the certain event.