1. WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam WFM-6204: Hydrologic Statistics Akm Saiful Islam Lecture-2: Probability and statistics December, 2006 Institute of Water and Flood Management (IWFM) Bangladesh University of Engineering and Technology (BUET) WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam

2. WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam Probability If a random event can occur in n equal likely and mutually exclusive ways, and if n a of these ways have an attribute A, then probability of the occurrence of the event having attribute A is, Prob(A) = n a /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

3. WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam Example-1: Find the probability of peak flow in excess of 1,00,000 cfs will occur. What should be the probability of this excess in 2 successive years ? Year190719171927193719571967 Flow66,300111,00093,700112,00088,700115,00

4. WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam Laws of probability 1. General Addition rule:  If A and B are two mutually inclusive events in S so that is not empty, then the probability of A or B is given by:  The notation represents a union so that represents all elements in A or B or both. The notation represents an intersection so that represents all elements in both A and B.

5. WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam Venn diagram

6. WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam  If and are mutually exclusive events, then both cannot occur and. In this case,  If represents all elements in that are not in, then

7. WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam 2. Multiplication rule:  If the probability of an event B depends on the occurrence of an event A, then we write and say the probability of B given that A has occurred. Thus the conditional probability (the ratio) is given by:  From this equation, we find that, where

8. WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam Occasionally, we must deal with more than two events. If A, B and C are three non-mutually exclusive events, then where The two events A and B are independent if

9. WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. 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?

10. WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam Probability theorems 1. Total probability theorem  If represents a set of mutually exclusive and collectively exhaustive events, one can determine the probability of another event A from Venn diagram for total probability theorem

11. WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. 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?

12. WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. Akm Saiful Islam 2. Bayes theorem Bayes theorem deals with conditional probability. It can be used to find 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:  Bayes theorem provides a means of estimating probabilities of one event by observing a second event.

13. WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. 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)Prob of 1000 or more visitorsProb of being in temp class <600.050.05 60-700.200.15 70-800.500.20 80-900.750.35 90-1000.500.20 >1000.250.05

14. WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. 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 is also a random variable.

15. WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. 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.

16. WFM 6204: Hydrologic Statistics © Dr. Akm Saiful IslamDr. 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:  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.