UGA CRSS 4500-6500 - ch4 (11 pages)

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ch4



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ch4

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11
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University of Georgia
Course:
Crss 4500-6500 - Introduction to Gene Technology
Introduction to Gene Technology Documents
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Chapter 4 Statistical Hydrology Statistics is the science of understanding uncertainty Will it rain today Given that it has not rained for three months what is the probability that it might rain in the next week How does a dam or ground water pumping wetland construction timber harvesting affect streamflow What are the health risks from drinking contaminated water These are all questions that are commonly asked questions that statistics might be able to help answer While the goal of science is to separate fact from fiction we are often limited to providing statistical measures of truth and error Thus science often rests on the edge of certainty not entirely sure yet nor entirely unsure Many of the early discoveries in statistics were made by compulsive gamblers who wished to improve their odds of winning Rather than accept the roll of the dice these individuals wanted to better understand the risks they were taking and place their bets in ways that maximized their likelihood of winning 4 1 We can also say that the probability of two events A and B occurring together is just the product of the probability of each event P A B P A and B P A P B 4 2 For example let us assume that the probability of landing either a heads H or a tails T when a coin is flipped are equal so that there are two outcomes with the probability of obtaining one or the other being p 0 5 When the coin is tossed twice n 2 there are four outcomes H H T T H T and T H Each outcome has an equal probability because these are independent events The probability of two heads in a row is P H H p2 0 25 4 3 which is the same for landing two tails The probability of landing one of each has two outcomes so that P T H H T P T H P H T Probability 2 p2 0 5 4 4 Probability analysis is used to describe random behavior We can write this mathematically for any number of tosses such as the chance that an event will occur or the likeli n to determine the number of heads m and tails n m hood that an event will exceed a certain magnitude While n much of nature is not entirely random we can often apply P H m T n m P H m P T n m m probability models to natural systems We can make these applications more readily in cases where n pn 4 5 Events are independent of each other and do not m affect each other That is the result of one coin toss where does not affect the following coin toss n n 4 6 Events are stationary they are not a function of m n m m time That is heads are not more likely in the morning than in the evening is the combinatorial operator that accounts for the number Events are identically distributed That is the vari of opportunities for getting the same outcome and where n n n 1 n 2 1 is the factorial of n ability of heads is the same under all conditions For example we can calculate the probability of getIf these assumptions are satisfied then we can say that ting exactly 5 heads and 5 tails in ten tosses the likelihood of either one event or another occurring is 5 5 1 10 1 just the sum of the individual events P H 5 T 5 2 2 5 P A B P A or B P A P B 4 1 1 2 CHAPTER 4 STATISTICAL HYDROLOGY Problems 1 For a daily rainfall probability of 30 percent find the probability that it will a Rain three days in a row b Not rain for seven days in a row c Rain three days in a week d Rain at least 1 day in a week e Rain greater than one inch if the probability is 25 percent if it rains Figure 4 1 Venn diagram of events A and B 10 10 1 0 246 2 5 Digression Bayes Theorem 4 7 Bayes theorem can be used to predict events by incorporating the relationship between events For example if Joe and Susan which means that we have a chance of only about 1 in 4 are frequently found together then if you see Susan you have a high likelihood of seeing Joe of getting an equal number of heads and tails The theorem is derived by noting that If events are not independent of each other then we can still calculate their probability using P A B P A B P A B P A B P A P B 4 8 4 13 P B P A B 4 14 P B A For example if the probability of a rainy day is thirty P A percent P A 0 3 the probability of snow is ten percent P B 0 1 and the probability of getting both rain If A and B are independent events then P A B P A P B so that and snow in a day is five percent P A B 0 05 then the probability of getting either rain or snow in a day is P A B P A P B P A B P A B 0 3 0 1 0 05 0 35 4 9 P B P B P A 4 15 and Frequency vs probability The frequency of an outP A B P A P B P B A P B 4 16 come is the observed number of occurrences based on a P A P A finite sample size fi ni N while a probability is the which makes sense because knowing A doesn t help knowing likelihood of that outcome based on an infinite sample size ni pi lim N N 4 10 We might say that the probability is the predicted frequency based on complete information Conditional probability Conditional probabilities are used when an event A may be more or less likely given that another event B has happened In this case P A P A B P B A P B 4 11 where P A B means the probability of Event A given that Event B has already occurred For example if the probability of rain and snow are again 30 and 10 percent respectively and the probability of snow given that rainfall has occurred is 20 percent then the probability of rain given that snow has occurred is 0 3 0 2 0 6 4 12 P A B 0 1 or sixty percent B and vice versa If on the other hand A and B are related to each other then we can use information about one to help with our prediction Combining the first two equations yields P A B P B A P A P A B P B 4 17 P A P B 4 18 or P A B P B A Example Say we have a relationship between temperature and snowfall Let P …


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