The most important birthday of my life is my 11th birthday .I remember as its yesterday. It was a beautiful Saturday morning when I woke up due to a loudly noise. It was 8:00 am then I knew that it was October 12, my birthday. I took a long, cold shower to calm myself down and got dressed. I was too excited. I was still in bed when all of the sudden I started to hear noise at the outside of my room. I went down stairs to the kitchen. Just as I entered the kitchen I became aware of a delicious
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theory. Probability also plays a key role in fields such as Physics, Biology, Economics and Medicine. There is a close relationship between Counting/Combinatorics and Probability. In many cases, the probability of an event is simply the fraction of possible outcomes that make up the event. So many of the rules we developed for finding the cardinality of finite sets carry over to Probability Theory. For example, we’ll apply an Inclusion-Exclusion principle for probabilities in some examples below. In principle
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There are three elements of a probability which combine to equal a result. There is the experiment ,the sample space and the event (Editorial board, 2012). In this case the class is the experiment because the process of attempting it will result in a grade which could vary from an A to F. The different grades that can be achieved in the class are the sample space. The event or outcome is the grade that will be received at the end of the experiment. I would like to achieve an “A” in this class but
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uncertain event will occur (always between 0 and 1) Impossible Event – an event that has no chance of occurring (probability = 0) Certain Event – an event that is sure to occur (probability = 1) Assessing Probability probability of occurrence= probability of occurrence based on a combination of an individual’s past experience, personal opinion, and analysis of a particular situation Events Simple event An event described by a single characteristic Joint event An event described
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or countable infinite set of outcomes • Out of the previous examples, which ones are discrete sample spaces??? Continuous sample space: One that contains an interval of real numbers. The interval can be either finite or infinite 5 Events A collection of certain sample points A subset of the sample space Denoted by ‘E’ Examples: • Getting an odd number in dice throwing experiment S = {1, 2, 3, 4,
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covariance, correlation. Limiting distributions. The syllabus is as follows: 1. Basic notions of probability. Sample spaces, events, relative frequency, probability axioms. 2. Finite sample spaces. Methods of enumeration. Combinatorial probability. 3. Conditional probability. Theorem of total probability. Bayes theorem. 4. Independence of two events. Mutual independence of n events. Sampling with and without replacement. 5. Random variables. Univariate distributions - discrete, continuous, mixed. Standard
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[pic] [pic] Markov Chain [pic] Bonus Malus Model [pic] [pic] This table justifies the matrix above: | | | |Next state | | | |State |Premium |0 Claims |1 Claim
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Permutations The word ‘coincidence’ is defined as an event that might have been arranged though it was accidental in actuality. Most of us perceive life as a set of coincidences that lead us to pre-destined conclusions despite believing in a being who is free from the shackles of time and space. The question is that a being, for whom time and space would be nothing more than two more dimensions, wouldn’t it be rather disparaging to throw events out randomly and witness how the history unfolds (as
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space for the radar detection problem Solution: Let A and B be the events A={an aircraft is present}, B={the radar registers an aircraft presence}, and consider also their complements Ac={an aircraft is not present}, Bc={the radar does not register an aircraft presence}. The given probabilities are recorded along the corresponding branches of the tree describing the sample space, as shown in Fig. 1. Each event of interest corresponds to a leaf of the tree and its probability is equal
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1.M/G/ Queue a. Show that Let A(t) : Number of arrivals between time (0, t] “ n should be equal to or great than k” since if n is less than k (n<k), Pk(t)=0 Let’s think some customer C, Let’s find P{C arrived at time x and in service at time t | x=(0,t)] } P{C arrives in (x, x+dx] | C arrives in (0, t] }P{C is in service | C arrives at x, and x = (0,t] } Since theorem of Poisson Process, The theorem is that Given that N(t) =n, the n arrival times S1, S2
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