Geometric Distribution

- 1 - CHƯƠNG I GIỚI THIỆU VỀ CÔNG FPT DISTRIBUTION 1.1.Qúa trình hình thành và phát triển của FPT DISTRIBUTION COMPANY (FDC): Công ty Phân phối FPT đã và đang dẫn đầu thị trường công nghệ thông tin (CNTT) và viễn thông tại Việt Nam, luôn sát cánh cùng các đối tác và hệ thống đại lý của mình mang đến cho người tiêu dùng Việt Nam những sản phẩm CNTT mới nhất, đa dạng về chủng loại và hoàn hảo về chất lượng. Là một công ty thành viên của Tập đoàn FPT, được chính thức thành lập từ ngày 13/4/2003 với trụ sở chính đặt tại Hà Nội và chi nhánh ở Tp. Hồ Chí Minh, Đà Nẵng, Cần Thơ, công ty luôn tự hào là đơn vị có thành tích kinh doanh nổi bật trong tập đoàn, với doanh thu năm 2007 vượt mức 516 triệu USD và tốc độ tăng trưởng hàng năm đạt hơn 59%. Hiệu quả hoạt động của Công ty Phân Phối FPT đã được khẳng định bởi chứng chỉ hệ thống quản lý chất lượng đạt tiêu chuẩn ISO9001:2000. Công ty Phân phối FPT đã chứng minh được vị thế số 1 trong lĩnh vực phân phối các sản phẩm CNTT và Viễn thông trên thị trường Việt Nam. Công ty Phân phối FPT có cơ cấu tổ chức chặt chẽ và thống nhất trên toàn quốc với đội ngũ nhân viên đông đảo, nhiệt tình, năng động, sáng tạo, có trình độ chuyên môn và năng suất lao động cao, trong đó trên 92% số nhân viên có kinh nghiệm hoạt động trong lĩnh vực CNTT, viễn thông và phân phối. Hệ thống thông tin đóng vai trò hết sức quan trọng trong thành công của Công ty, trong đó phải kể đến hệ thống thông tin tài chính và thông tin quản lý: phần mềm kế toán Oracle, FIFA (FPT Information Finance Architecture), MIS (Management Information System), SCM (Supply Chain Management), CRM (Customer Relationship Management), HRM (Human Resource Management), FDC Inside . Với những thế mạnh sẵn có cùng tôn chỉ hướng tới khách hàng, Công ty Phân phối FPT cam kết tiếp tục mang đến cho khách hàng của mình những giá trị gia tăng, giữ vững niềm tin và uy tín với các đối tác, tiếp tục đứng vững trên thị trường trong nước và vươn ra thị trường nước ngoài -Tên giao dịch : FPT Distribution -Ngày thành lập : 13/04/2003 (kết hợp từ 3 trung tâm phân phối của FPT) -Trụ sở : 298G Kim Mã, Quận Ba Đình, Hà Nội -Vốn đầu tư : 516 triệu USD Geometric Distribution Geometric Distribution By: OpenStaxCollege There are three main characteristics of a geometric experiment There are one or more Bernoulli trials with all failures except the last one, which is a success In other words, you keep repeating what you are doing until the first success Then you stop For example, you throw a dart at a bullseye until you hit the bullseye The first time you hit the bullseye is a "success" so you stop throwing the dart It might take six tries until you hit the bullseye You can think of the trials as failure, failure, failure, failure, failure, success, STOP In theory, the number of trials could go on forever There must be at least one trial The probability, p, of a success and the probability, q, of a failure is the same for each trial p + q = and q = − p For example, the probability of rolling a three when you throw one fair die is This is true no matter how many times you roll the die Suppose you want to know the probability of getting the first three on the fifth roll On rolls one through four, you not get a face with a three The probability for each of the rolls is q = , the probability of a failure The probability of getting a three on the fifth roll is ( 56 )( 56 )( 56 )( 56 )( 16 ) = 0.0804 X = the number of independent trials until the first success You play a game of chance that you can either win or lose (there are no other possibilities) until you lose Your probability of losing is p = 0.57 What is the probability that it takes five games until you lose? Let X = the number of games you play until you lose (includes the losing game) Then X takes on the values 1, 2, 3, (could go on indefinitely) The probability question is P(x = 5) Try It You throw darts at a board until you hit the center area Your probability of hitting the center area is p = 0.17 You want to find the probability that it takes eight throws until you hit the center What values does X take on? 1, 2, 3, 4, … n It can go on indefinitely 1/11 Geometric Distribution A safety engineer feels that 35% of all industrial accidents in her plant are caused by failure of employees to follow instructions She decides to look at the accident reports (selected randomly and replaced in the pile after reading) until she finds one that shows an accident caused by failure of employees to follow instructions On average, how many reports would the safety engineer expect to look at until she finds a report showing an accident caused by employee failure to follow instructions? What is the probability that the safety engineer will have to examine at least three reports until she finds a report showing an accident caused by employee failure to follow instructions? Let X = the number of accidents the safety engineer must examine until she finds a report showing an accident caused by employee failure to follow instructions X takes on the values 1, 2, 3, The first question asks you to find the expected value or the mean The second question asks you to find P(x ≥ 3) ("At least" translates to a "greater than or equal to" symbol) Try It An instructor feels that 15% of students get below a C on their final exam She decides to look at final exams (selected randomly and replaced in the pile after reading) until she finds one that shows a grade below a C We want to know the probability that the instructor will have to examine at least ten exams until she finds one with a grade below a C What is the probability question stated mathematically? P(x ≥ 10) Suppose that you are looking for a student at your college who lives within five miles of you You know that 55% of the 25,000 students live within five miles of you You randomly contact students from the college until one says he or she lives within five miles of you What is the probability that you need to contact four people? This is a geometric problem because you may have a number of failures before you have the one success you desire Also, the probability of a success stays the same each time you ask a student if he or she lives within five miles of you There is no definite number of trials (number of times you ask a student) a Let X = the number of you must ask one says yes a Let X = the number of students you must ask until one says yes b What values does X take on? b 1, 2, 3, …, (total number of students) c What are p and q? 2/11 Geometric Distribution c p = 0.55; q = 0.45 d The probability question is P( _) d P(x = 4) Try It You need to find a store that carries a special printer ink You know that of the stores that carry printer ink, 10% of them carry the special ink You randomly call each store until one has the ink you need What are p and q? p = 0.1 q = 0.9 Notation for the Geometric: G = Geometric Probability Distribution Function X ~ G(p) Read this as "X is a random variable with a geometric distribution." The parameter is p; p = the probability of a success for each trial Assume that the probability of a defective computer component is 0.02 ... - 67 - CHARACTERISTICS AND DISTRIBUTIONS OF NITROUS OXIDE-PRODUCING DENITRI- FYING FUNGI IN NATURAL ENVIRONMENTS K. Oishi and T. Kusuda Department of Civil Engineering, Faculty of Engineering, Kyushu University, Hakozaki 6-10-1, Higashi-ku, Fukuoka 812-8581, Japan Abstract Tea field soils, and sediments of an irrigation pond and a tidal river, in which a variety of organic matter was supplied as energy sources, were collected. The activities of bacterial and fungal denitrifications in these samples were determined. Denitrifying fungi in all of these samples produced N 2 O from nitrate and nitrite as a final product, whereas denitrifying bacteria produced N 2 . Nitrous oxide produced by fungi was reduced to N 2 by bacteria. The fungal denitrification potentials were the highest in the submerged litter on the pond sediment, followed by the farmyard manure-amended soil, the inorganic fertilizer-amended soil, the litter-free pond sediment, and the tidal river sediment. The enrichments of denitrifying fungi in natural envi- ronments were related with the distributions of the organic material such as straws and litter. The contributions of fungal denitrification to total denitrification were large in soil environments, especially in the farmyard manure-amended soil, and were small in aquatic environments such as the sediments of pond and river. The pH in situ was not related with the fungal denitrification potentials. Keywords fungal denitrification; bacterial denitrification; nitrous oxide; organic matter; sediments; soils Introduction Fungi generally are found in lakes, ponds, rivers, estuaries, marine, wastewater, and soils. Despite their wide occurrence, little attention has been given to the presence and ecological significance of fungi. Especially, characteristics and contributions of fungal denitrification in natural environments are poorly understood. Deni- trification is a process in which nitrite and/or nitrate are reduced to N 2 gas through N 2 O. The process has been considered to be mainly caused by bacteria. However, pure culture experiments have shown that fungi such as Fusarium sp., Trichoderma hamatum, Chaetomium sp., Gibberella fujikuroi etc., can reduce nitrite and several strains can reduce nitrate as well, but the final product is mainly N 2 O rather than N 2 ( Bleakley and Tiedje, 1982; Burth and Ottow, 1983; Shoun and Tanimoto, 1991; Shoun et al., 1992). The distributions of denitrifying fungi, which produce N 2 O as a final product, would be ecologically important to understand the contribution of natural systems to the atmospheric concentration of N 2 O. Previous studies have been con- ducted with pure cultures of fungi. However, the characteristics and distributions of denitrifying fungi in natural environments are unknown. In this study, tea soils, and the sediments of an irrigation pond and a tidal river, to which different types of organic matter were supplied, were collected. The final products of fungal denitrification in these soils and sediments were determined, and the distribution of denitrifying fungi in natural system was estimated. Materials Tea field soils Soils were collected at the surface (0-10cm) in two tea fields. One was mainly amended with a farmyard manure (organic soil), and the other with inorganic fertilizer Journal of Water and Environment Technology, Vol.3, No.2, 2005 - 169 - INFLUENCE DISTRIBUTIONS OF ACID DEPOSITION IN MOUNTAINOUS STREAMS ON A TALL CONE-SHAPED ISLAND, YAKUSHIMA S. Ebise* and O. Nagafuchi** *Dept. of Civil and Environmental Engineering System, Setsunan University, 17-8 Ikeda-Nakamachi, Neyagawa, Osaka, 572-8508, Japan **Dept. of Environmental System, Chiba Institute of Science, 3 Shiomi, Choshi, Chiba 288-0025 ABSTRACT Yakushima, facing at 800 km east of Shanghai in the East China Sea, is a tall cone-shaped island with seven exceeding 1800 m peaks. The prevailing winds of westrelie on the island blow mostly fromnorthwest and west. It has been exposed to acid rain of pH 4.7 and precipitation 8000 mm in the central highland. More than sixty mountainous streams were observed at downstream points seasonally for past twelve years. The alkalinity of streamwaters in the southwestern part was lower than others. The concentrations of SO 4 2- in the northwestern part were higher than others. The high concentrations of SO 4 2- , dissolved SiO 2 and other ions in the southwestern part with high canopy density of evergreen broadleaved forest were caused by higher air temperature, less rainfall and higher evapotranspiration than other parts. The alkalinity, pH and EC in the catchment of north stream in the upstream branch of the R. Anboh became lower than those in the catchment of south stream. The height of catchment boundaries, the direction of the main axis of a catchment and the prevailing winds govern the influence of acid deposition on water quality of branch streams. KEYWORDS Acid deposition; mountainous stream; influence distribution; cone-shaped; prevailing wind INTRODUCTION Yakushima, an island lying 800 km east of Shanghai on the boundary between the Pacific Ocean and East China Sea (Fig. 1), is a World Heritages Area and a National Park of Japan. Yakushima is famous for its yaku-sugi (Crytomeria japonica), one of which, called Jomon-sugi, is the oldest living organism in Japan. The prevailing winds on the island were northwestern (strongly so in winter) and western except during the short typhoon season, when the wind comes from the southeast. Consequently, Yakushima is exposed to acid deposition with an annual mean pH of 4.7 (MOE, 2004). The annual precipitation ranges 4300 mm at the coast to above 8000 mm in the central highland where seven peaks exceeds 1800 m. The tallest peak is Mt. Miyanouradake at 1935 m. The island has a steep mountainous landform and is covered with a thin soil layers overlying granite. Therefore, the soil’s ability to neutralize acid deposition is very weak (Ebise, 1996). Journal of Water and Environment Technology, Vol.3, No.2, 2005 - 170 - Figure 1 Yakushima Island and main rivers All 14,000 residents of the island live on the coast. There are no sources of artificial pollutants in the mountains, where the only visitors are mountain climber and backpackers. Back-trajectory and Pb-isotope trace methods show that most of the acid deposition coming from the Asia Continent is transported overseas by the prevailing westerlies in the upper air region, until it strikes the high mountains on the island. The major species in the acid deposition is sulfuric acid (Nagafuchi et al. 1995, 2001a, 2001b). The pH of wet deposition tends to be lower in Journal of Water and Environment Technology, Vol.4, No.1, 2006 - 61 - Effect of urban emissions on the horizontal distribution of metal concentration in sediments in the vicinity of Asian large cities T. Urase 1* , K. Nadaoka 2 , H. Yagi 2 , T. Iwasa 1 , Y. Suzuki 1 F. Siringan 3 , T. P. Garcia 4 , T. T. Thao 5 1: Dept. of Civil Engineering, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro, Tokyo, 152-8552 Japan. *: Corresponding author. turase@fluid.cv.titech.ac.jp , +81-3-5734-3548 2: Dept. of Mechanical and Environmental Informatics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro, Tokyo, 152-8552 Japan. 3: National Institute of Geological Sciences, University of the Philippines, 1101 Diliman, Quezon City, Philippines 4: Dept. Civil Engineering, College of Engineering, Technological University of the Philippines, Manila, Philippines 5: Department of Analytical Chemistry, Hanoi University of Science, 19- LeThanh Tong street, Hanoi, Vietnam Abstract: Metal contents of sediments in Manila Bay – Laguna Lake watershed in the Philippines were measured and detailed horizontal distribution was obtained. The distribution of zinc and lead concentration in Manila Bay clearly shows the effect of anthropogenic contamination and it was explained by the diffusion of lead and zinc rich anthropogenic particles discharged from Pasig River. The sediments in Laguna Lake were mostly natural particulate matters from surrounding mountains and they contained 20 mgPb/kg and 100 mgZn/kg, while the sediment taken at the heavily polluted branches of the Pasig River contained as high as 88 mgPb/kg and 310 mgZn/kg. The lead and zinc concentrations in the sediments of Manila Bay – Laguna Lake watershed were compared with those in the mouth of the Tama River, Tokyo, where the faster deposition of coarser natural origin particles and slower deposition of lead and zinc rich anthropogenic particles determined the sediment concentration. The comparison was also made with Hanoi City, Vietnam. In spite of the difference in time when leaded gasoline was prohibited, the difference in the lead concentrations of roadside deposits and sediments was not obvious in the vicinity of these three target cities. This is probably due to dilution by a large amount of suspended solids conveyed by the Pasig River in the case of the Philippines. Storm water runoff containing roadside deposits and discharge of untreated wastewater were identified as factors increasing zinc and lead concentrations of sediments in receiving waters based on the measurements on roadside deposits and the estimation of the contribution of untreated wastewater. Keywords: Laguna Lake; lead; Manila Bay; sediment; wastewater; zinc. Introduction Asian cities generally have large populations. Human activities and their impacts on natural environments are concentrated in the vicinity of urban regions. High precipitation in Asian regions results in erosion of land and induces urban runoff during wet weather days. A large amount of particulate matters having natural and anthropogenic sources flows into receiving watersheds. Incomplete sewer problems such as low coverage and Probability in Computing © 2010, Quoc Le & Van Nguyen Probability for Computing 1 LECTURE 4: CONDITIONAL PROBABILITY, APPLICATIONS, GEOMETRIC DISTRIBUTION Agenda Application: Verification of Matrix Multiplication Application: Randomized Min - Cut © 2010, Quoc Le & Van Nguyen Probability for Computing 2 Application: Randomized Min - Cut Geometric Distribution Coupon Collector’s Problem Application: Verifying Matrix Multiplication Consider matrix multiplication AB = C (integers modulo 2)  Simple algorithm takes O(n 3 ) operations.  Want to check if a given matrix multiplication program works correctly Randomized Algorithm: © 2010, Quoc Le & Van Nguyen Probability for Computing 3 Randomized Algorithm:  Choose a random vector r = (r 1 , r 2 , …, r n ) in {0,1} n .  Compute A(Br) and Cr then comparer the two values: if equal return yes AB=C, else no. Note on the above randomized algorithm:  1-side error  Complexity = O(n 2 )  Accuracy depends on P(ABr = Cr) when AB!=C Analysis of P(ABr = Cr) Choosing r randomly is equivalent to choosing r i randomly and independently. (1) Let D = AB – C ≠ 0. Since Dr = 0, there must be some non-zero entry. Let that be d 11 .  Dr = 0  ∑d 1j r j = 0  r 1 = -∑d 1j r j / d 11 .  Since r1 can take 2 values, combine with (1), we have ABr = Cr with © 2010, Quoc Le & Van Nguyen Probability for Computing 4  Since r1 can take 2 values, combine with (1), we have ABr = Cr with probability of at most ½  Refer to book for formal proof (using Law of Total Probability) Principle of Deferred Decisions: when there are several random variables, it often helps to think of some of them as being set at one point in the algorithm with the rest of them being left random (or deferred) until some further point in the analysis. We can attempt this verification k times to obtain accurate answer with p = 2 -k and efficiency = O(kn 2 ) = O(n 2 ) Theorems Law of Total Probability: Assume E 1 , E 2 , …, E n be mutually disjoint events in the sample space Ω and union of E i = Ω. Then  Pr(B) = ∑Pr(B and E i ) = ∑Pr(B|E i )Pr(E i ) Bayes’ Law: Assume E , E , …, E be mutually disjoint © 2010, Quoc Le & Van Nguyen Probability for Computing 5 Bayes’ Law: Assume E 1 , E 2 , …, E n be mutually disjoint events in the sample space Ω and union of E i = Ω. Then Pr(E j |B) = Pr(E j and B)/Pr(B) = Pr(B|E j )Pr(E j ) / ∑Pr(B|E i )Pr(E i )  Notice the model transformation from prior probability to posterior probability. Gradual Change in Our Confidence in Algorithm Correctness In matrix verification case:  E = the identify is correct  B = test returns that the identity is correct Prior assumption: Identity = ½  How does this assumption change after each run? © 2010, Quoc Le & Van Nguyen Probability for Computing 6  How does this assumption change after each run? We start with Pr(E) = Pr(E c ) = ½ Since the test has error bounded by ½, Pr(B|E c ) ≤ ½. Also, Pr(B|E) = 1 Now by Bayes’ Law: Pr(E|B) = Pr(B|E)Pr(E) / {Pr(B|E)Pr(E)+ Pr(B|E c )Pr(E c )}≥ ½ / {1.½ + ½. ½} = 2/3 The prior model is revised:  Pr(E) ≥ 2/3 and Pr(E c ) ≤ 1/3. Applying Bayes’ Law again will yeild Pr(E|B) ≥ 4/5 Gradual Change in Our Confidence in Algorithm Correctness © 2010, Quoc Le & Van Nguyen Probability for Computing 7 4/5 In general, at i th iteration, Pr(E|B) ≥1 – 1/(2 i +1) After 100 calls, test returns that identity is correct, then our confidence in the correctness of this identity is at least 1 – 1/ 2 100 +1) Application: Randomized Min Cut Cut-set: Set of edges whose removal breaks the graph into two or more connected components. © 2010, Quoc Le & Van Nguyen Probability for Computing 8 Min-cut: Cut-set with minimum cardinality. Applications:  Network reliability.  Clustering problems  Al-Qaeda Example A B G E F © 2010, Quoc Le & Van Nguyen Probability for Computing 9 C B D G E Example A B G E F © 2010, Quoc Le & Van Nguyen Probability for Computing 10 C B D G E E1 = BE ... the geometric distribution X ~ G(p) is μ = √ 1−p p2 = √( 1 p p ) −1 6/11 Geometric Distribution Formula Review X ~ G(p) means that the discrete random variable X has a geometric probability distribution. .. q? p = 0.1 q = 0.9 Notation for the Geometric: G = Geometric Probability Distribution Function X ~ G(p) Read this as "X is a random variable with a geometric distribution. " The parameter is p;... for each trial In a geometric experiment, define the discrete random variable X as the number of independent trials until the first success We say that X has a geometric distribution and write