Chebyshev’s Theorem and The Empirical Rule Suppose we ask 1000 people what their age is. If this is a representative sample then there will be very few people of 1-2 years old just as there will not be many 95 year olds. Most will have an age somewhere in their 30’s or 40’s. A list of the number of people of a certain age may look like this: |Age |Number of people | |0 |1 | |1 |2 | |2 |3
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The four color theorem is a mathematical theorem that states that‚ given a map‚ no more than four colors are required to color the regions of the map‚ so that no 2 regions that are touching (share a common boundary) have the same color. This theorem was proven by Kenneth Appel and Wolfgang Haken in 1976‚ and is unique because it was the first major theorem to be proven using a computer. This proof was first proposed in 1852 by Francis Guthrie when he was coloring the counties of England and realized
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1 10/10/01 Fermat’s Little Theorem From the Multinomial Theorem Thomas J. Osler (osler@rowan.edu) Rowan University‚ Glassboro‚ NJ 08028 Fermat’s Little Theorem [1] states that n p −1 − 1 is divisible by p whenever p is prime and n is an integer not divisible by p. This theorem is used in many of the simpler tests for primality. The so-called multinomial theorem (described in [2]) gives the expansion of a multinomial to an integer power p > 0‚ (a1 + a2 + ⋅⋅⋅ + an ) p = p k1 k2 kn a1 a2
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Kirchhoff’s Law Kirchhoff’s current law (KCL) imposes constraints on the currents in the branches that are attached to each node of a circuit. In simplest terms‚ KCL states that the sum of the currents that are entering a given node must equal the sum of the currents that are leaving the node. Thus‚ the set of currents in branches attached to a given node can be partitioned into two groups whose orientation is away from (into) the node. The two groups must contain the same net current. In general
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“The Arrow impossibility theorem and its implications for voting and elections” Arrow’s impossibility theorem represents a fascinating problem in the philosophy of economics‚ widely discussed for insinuating doubt on commonly accepted beliefs towards collective decision making procedures. This essay will introduce its fundamental assumptions‚ explain its meaning‚ explore some of the solutions available to escape its predictions and finally discuss its implications for political
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Limit at infinity The function has the limit as increases without bound (or‚ asapproaches infinity)‚ written if can be made arbitrarily close to by taking large enough. Similarly‚ the function has the limit as decreases without bound (or‚ asapproaches negative infinity)‚ written if can be made arbitrarily close to by taking to be negative and sufficiently large in absolute value. One-sided limits The function has the right-hand limit as approaches from the right written if the
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mathematics that developed from simple measurements. A theorem is the most important result in all of elementary mathematics. It was the motivation for a wealth of advanced mathematics‚ such as Fermat’s Last Theorem and the theory of Hilbert space. The Pythagorean Theorem asserts that for a right triangle‚ the square of the hypotenuse is equal to the sum of the squares of the other two sides. There are many ways to prove the Pythagorean Theorem. A particularly simple one is the scaling relationship
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1. As the degrees of freedom increase‚ the t distribution approaches the b. normal distribution 2. If the margin of error in an interval estimate of μ is 4.6‚ the interval estimate equals b. [pic] 3. The t distribution is a family of similar probability distributions‚ with each individual distribution depending on a parameter known as the c. degrees of freedom 4. The probability that the interval estimation procedure will generate an interval that does
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Descriptive statistics information Descriptive statistics organize‚ summarize‚ and communicate a group of numerical observations and describe large amounts of data in a single number or in just a few numbers Inferential statistics Use samples to draw conclusions about a population Inferential statistical use sample data to make general estimates about the larger population‚ and infer or make an intelligence guess about‚ the population Sample: a set of observations drawn from the population
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a NOR gate. DeMorgan’s theorems state the same equivalence in "backward" form: that inverting the output of any gate results in the same function as the opposite type of gate (AND vs. OR) with inverted inputs De Morgan’s theorem is used to simplify a lot expression of complicated logic gates. For example‚ (A + (BC)’)’. The parentheses symbol is used in the example. _ The answer is A BC. Let’s apply the principles of DeMorgan’s theorems to the simplification of
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