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Introduction |
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In our day to day life, we come across many uncertainty of events. We wake up in the morning and check the weather report. The statement could be 'there is 60% chance of rain today'. This statement infers that the chance of rain is more than that having a dry weather. We decide upon our breakfast from a statement that "corn flakes might reduce cholesterol". What is the chance of getting a flat tyre on the way to an important apartment? And so on. |
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How probable an event is? We generally infer by repeated observation of such events in long term patterns. |
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Probability is the branch of mathematics devoted to the study of such events. |
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Random Experiment and Sample Space |
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An experiment repeated under essentially homogeneous and similar conditions results in an outcome, which is unique or not unique but may be one of the several possible outcomes. When the result is unique then the experiment is called a deterministic experiment. |
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Any experiment whose outcome cannot be predicted in advance, but is one of the set of possible outcomes, is called a random experiment. |
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If we think an experiment as being performed repeatedly, each repetition is called a trial. We observe an outcome for each trial. |
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The set of all possible outcomes of a random experiment is called the sample space, associated with the random experiment. |
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Events |
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An event is the outcome or a combination of outcomes of an experiment. In other words, an event is a subset of the sample space. |
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Whenever an outcome satisfies the conditions, given in the event, we say that the event has occurred. |
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The Types of Events are: Simple Event, Compound Event, Null Event, sure event or certain event, Complement of an Event, Algebra of Event, Mutually Exclusive Event, Exhaustive Event, Equally Likely Outcomes. |
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Probability of an Event |
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If a trial results in n-exhaustive, mutually exclusive and equally likely cases and m of them are favourable to the occurrence of an event A, then the probability of the happening of A, denoted by P(A), is given by:
P(A) = m/n.
Important terms are 1. Statistical or Empirical Probability, 2. Axiomatic Approach to Probability. |
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Theorems of Probability |
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1. Addition Rule of Probability: If A and B are any two events, then
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2. P(AC) = 1 - P(A). |
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3. P(f) = 0. |
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Multiplication Rule of Probability |
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Events are said to be independent if the occurrence of one event does not affect the occurrence of others. |
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If A and B are two independent events, then
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This is known as Multiplication Rule of Probability. The converse is also true, that is if two events A and B associated with a random experiment.
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Independent Experiment:
Two random experiments are said to be independent if for every pair of events E and F, where E is associated with the first experiment and F is associated with the second experiment, the probability of simultaneous occurrence of E and F, when the two experiments are performed, is the product of the probabilities P(E) and P(F).
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Random Variables and Probability Distributions |
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Let S be a sample space associated with a given random experiment. |
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A real valued function X which assigns to each wi
Î S, a unique real number, X(wi)
= xi is called a random variable. Two types of random variables are 1. Continuous random variable, 2. discrete random variable. |
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Let X be a discrete random variable which takes values x1, x2, x3,…xn where pi = P{X = xi} |
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Then X : x1 x2 x3 …xn |
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P(X) : p1 p2 p3 … pn is called the probability distribution of x. |
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Summary |
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1. Sample space: Set of all possible outcomes of a random experiment. |
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2. Event : An event of a random experiment is defined as a subset of the sample space. |
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3. Exhaustive outcomes: All the outcomes of a random experiment. |
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4. Random variable: A real valued function 'X' defined on the sample space is called a random variable. |
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5. Discrete random variable: A random variable which can assume only finitely or infinitely many distinct values. |
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6. Continuous random variable: A random variable which can take any value over an interval is called a continuous random variable.
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