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17 Cards in this Set
- Front
- Back
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Sample Space
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a sample space is the set of all outcomes from an experiment, written S
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Event
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an event is a subset of the sample space, E.
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Event Space
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The event space is the set of all events, E of s
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Probability
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A probability space is a function mapping the event space into the interval between 0 and one such that.
1) 0 is less than the probability of a. and the probability a. is less than or equal to one, for all a. in the event space. 2) the probability of the space is equal to 1. 3)if a. intersect b = the null space. then the probability of a. union b is equal to the probability of a. plus the probability of B |
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Probability space
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Let the sample space, event space, and probability. be a probability space. defined by one. the probability of the null space equals zero 2. the probability of a. complement equals one minus the probability of A. and 3. the probability of a. union B is equal to the probability of a. plus the probability of B minus the probability of a. union B
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Independence
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Let A. be B in the event space. we say that a. & B are independent if the probability of a. intersect b equals the probably of a. times the probability of B
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Conditional probability
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let a. and b be in the event space. the probability of a. is greater than 0. we define the conditional probability of b given a as the probability of b given a. is equal to the probability of B intersect a. divided by the probability of a.
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Increasing set
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let a. sub i be in the event space, and a. sub i be a Sequence of events, with A. sub i as a subset of A. sub i plus 1, in this case we say the sequence is increasing
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Borel space
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The borel space is the class of sets derived from the results of all possible countable unions and intersections of the open and closed intervals of A. & B where, A. & B belong to the real set of numbers
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The standard measure
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The standard measure on the real numbers is defined by Sigma a. and b is equal to b minus a. this is called the length.
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Probability measure
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If u is a measure on the real space, then u of the real space equals one. we say this is a probability measure
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Random variable
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Let Omega, the event space of a mega and P be a probability measure space. we say that x is a random variable if one. X maps Omega on to the real space 2. if X inverse of A. is in the event space of omega for all A. in the borel space and 3. the probability of the set of omega such that the function X of omega equals plus or minus infinity equals 0.
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Probability Distribution function
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the probability Distribution function big F of big X of x associated with a random variable big X on the sample space, the event space, and the probability space is a function defined by, big F of big X of x equals, the probability of big X less than or equal to little x, which is equal to the probability of the set omega, such that big X of omega is less than or equal to little x, which is equal to the probability of big X on the interval between minus infinity to little x.
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unit step function
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the unit step function equals I for x greater than 0, and 0 for x less than or equal to 0.
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probability density function
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the probability density function little f of big X of x is equal to the derivative of Big F of x.
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Delta Function
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the delta Function is a generalized function that one, the integral of the delta Function from minus infinity to x is equal to the unit step function of x. and two. the integral from minus infinity to infinity of f of x minus y times the delta Function of y D. Y. equals f of x , for all f with f of plus or minus infinity equal to 0.
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uniform distribution
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for b greater than a., big F of x equals, 0 for x less than a., x minus a. divided by b minus a. for a
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