Slides Chapter 3. Laws of large numbers
... P Xi → p. n i=1 The next theorem does not require the existence of the variances, but in turn requires the r.v.s to be identically distributed. Theorem 3.2 (Khintchine’s weak law of large numbers) Let {Xn}n∈IN be a sequence of i.i.d. r.v.s with mean E(Xn) = µ ∈ (−∞, ∞). Then, n 1X P Xi → µ. n i=1 ...
... P Xi → p. n i=1 The next theorem does not require the existence of the variances, but in turn requires the r.v.s to be identically distributed. Theorem 3.2 (Khintchine’s weak law of large numbers) Let {Xn}n∈IN be a sequence of i.i.d. r.v.s with mean E(Xn) = µ ∈ (−∞, ∞). Then, n 1X P Xi → µ. n i=1 ...
Section 6.1 Class Notes
... To find the mean (expected value) of X, multiply each possible value by its probability then add all the products: ...
... To find the mean (expected value) of X, multiply each possible value by its probability then add all the products: ...
Some Conditions may apply
... The key notion here is that of admissibility. We can’t allow arbitrary evidence to be included, since some E’s will make the advice the PP gives incorrect. Admissible evidence includes Historical information [e.g., past observations about this coin] Theoretical information about the dependence o ...
... The key notion here is that of admissibility. We can’t allow arbitrary evidence to be included, since some E’s will make the advice the PP gives incorrect. Admissible evidence includes Historical information [e.g., past observations about this coin] Theoretical information about the dependence o ...
Introduction to probability (4)
... 7 or 8 or more cars on any given workday are respectively: 0.12, 0.19, 0.28, 0.24, 0.10 and 0.07. What is the probability that it will be serve at least 5 cars on next day at work. • Solution: Let E be the event that at least 5 cars are served ...
... 7 or 8 or more cars on any given workday are respectively: 0.12, 0.19, 0.28, 0.24, 0.10 and 0.07. What is the probability that it will be serve at least 5 cars on next day at work. • Solution: Let E be the event that at least 5 cars are served ...
PROBABILITY POSSIBLE OUTCOMES
... decimal. Since the number of ways a certain outcome may occur is always smaller or equal to the total number of outcomes, the probability of an event is some number from 0 through 1. Example: Suppose there are 10 balls in a bucket numbered as follows: 1, 1, 2, 3, 4, 4, 4, 5, 6, and 6. A single ball ...
... decimal. Since the number of ways a certain outcome may occur is always smaller or equal to the total number of outcomes, the probability of an event is some number from 0 through 1. Example: Suppose there are 10 balls in a bucket numbered as follows: 1, 1, 2, 3, 4, 4, 4, 5, 6, and 6. A single ball ...
Conditional Probability and Independence
... 1. Human Chorionic Gonadotropin (hCG) pregnancy tests or home pregnancy tests determine pregnancy through the detection of the hormone hCG in a woman’s urine. Tests too soon after conception can result in false negatives and certain aspects of recent health history of a women can result in false pos ...
... 1. Human Chorionic Gonadotropin (hCG) pregnancy tests or home pregnancy tests determine pregnancy through the detection of the hormone hCG in a woman’s urine. Tests too soon after conception can result in false negatives and certain aspects of recent health history of a women can result in false pos ...
return interval - University of Colorado Boulder
... of events; expressed in the unit time period): Tr = years of observation / # of events • Probability in any time period (# events / years, expressed in decimal or percent): p = # events/years – Percent can also be calculated as: p% = 100/r Here’s an example of a distribution of maximum annual stream ...
... of events; expressed in the unit time period): Tr = years of observation / # of events • Probability in any time period (# events / years, expressed in decimal or percent): p = # events/years – Percent can also be calculated as: p% = 100/r Here’s an example of a distribution of maximum annual stream ...
1) Randomization in algorithms.
... The PCP theorem: the AMAZING POWER OF PROBABILITY A prover sends (in a certain special form) a proof that some input x belongs to an NPC language L The prover looks only at randomly chosen CONSTANT number of bits from the proof!! Uses only log n randomization. If xL the verifier will say y ...
... The PCP theorem: the AMAZING POWER OF PROBABILITY A prover sends (in a certain special form) a proof that some input x belongs to an NPC language L The prover looks only at randomly chosen CONSTANT number of bits from the proof!! Uses only log n randomization. If xL the verifier will say y ...
Probability - Cornell Computer Science
... by the outcomes of the coin tosses, leads to an accept state. Formally, we define a probabilistic Turing machine to be an ordinary deterministic TM with an extra semi-infinite read-only tape containing a binary string called the random bits. The machine runs as an ordinary deterministic TM, consulting ...
... by the outcomes of the coin tosses, leads to an accept state. Formally, we define a probabilistic Turing machine to be an ordinary deterministic TM with an extra semi-infinite read-only tape containing a binary string called the random bits. The machine runs as an ordinary deterministic TM, consulting ...
Question paper
... An engineer measures, to the nearest cm, the lengths of metal rods. (a) Suggest a suitable model to represent the difference between the true lengths and the measured lengths. ...
... An engineer measures, to the nearest cm, the lengths of metal rods. (a) Suggest a suitable model to represent the difference between the true lengths and the measured lengths. ...
Infinite monkey theorem
The infinite monkey theorem states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type a given text, such as the complete works of William Shakespeare.In this context, ""almost surely"" is a mathematical term with a precise meaning, and the ""monkey"" is not an actual monkey, but a metaphor for an abstract device that produces an endless random sequence of letters and symbols. One of the earliest instances of the use of the ""monkey metaphor"" is that of French mathematician Émile Borel in 1913, but the first instance may be even earlier. The relevance of the theorem is questionable—the probability of a universe full of monkeys typing a complete work such as Shakespeare's Hamlet is so tiny that the chance of it occurring during a period of time hundreds of thousands of orders of magnitude longer than the age of the universe is extremely low (but technically not zero). It should also be noted that real monkeys don't produce uniformly random output, which means that an actual monkey hitting keys for an infinite amount of time has no statistical certainty of ever producing any given text.Variants of the theorem include multiple and even infinitely many typists, and the target text varies between an entire library and a single sentence. The history of these statements can be traced back to Aristotle's On Generation and Corruption and Cicero's De natura deorum (On the Nature of the Gods), through Blaise Pascal and Jonathan Swift, and finally to modern statements with their iconic simians and typewriters. In the early 20th century, Émile Borel and Arthur Eddington used the theorem to illustrate the timescales implicit in the foundations of statistical mechanics.