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Yes it can.
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The moment generating function for any real valued probability distribution is the expected value of e^tX provided that the expectation exists.
For the Type I Pareto distribution with tail index a, this is
a*[-x(m)t)^a*Gamma[-a, -x(m)t)] for t < 0, where x(m) is the scale parameter and represents the least possible positive value of X.
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A pentillion was the name proposed for what is not known as a quadrillion = 10^15.
This was necessitated by the confusion between the US and European use of billion (thousand million or million million) and higher numbers. New names, based on Greek roots, were proposed. In many countries the European names have now been dropped in favour of the US names.
So the next term would have been a hextillion.
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Conjuncture is basically a joining of objects words or phrases. So to make a conjuncture would be to join things together. In propositional calculus making a conjuncture would mean joining a string of statements together.
Example: given the two statements
IF All A's are letters;
And All letters are are in the alphabet;
then the Conjuncture would be
All A's are in the alphabet.
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Best example is that an "odd" (or "even") function's Maclaurin series only has terms with odd (or even) powers. cos(x) and sin(x) are examples of odd and even functions with easy to calculate Maclaurin series.
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The probability distribution function (pdf) is defined over a domain which contains at least one interval in which the pdf is positive for all values.
Usually the domain is either the whole of the real numbers or the positive real numbers, but it can be a finite interval: for example, the uniform continuous distribution.
Also, trivially,
the pdf is always non-negative,
the integral of the pdf, over the whole real line, equals 1.
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Euler's formula states that eix = cos(x) + isin(x) where i is the imaginary number and x is any real number.
First, we get the power series of eix using the formula:
ez = Σ∞n=o zn/n! where z = ix. That gives us:
1 + ix + (ix)2/2! + (ix)3/3! + (ix)4/4! + (ix)5/5! + (ix)6/6! + (ix)7/7! + (ix)8/8! + ...
which from the properties of i equals:
1 + ix - x2/2! - ix3/3! + x4/4! + ix5/5! - x6/6! - ix7/7! + x8/8! + ...
which equals:
(1 - x2/2! + x4/4! - x6/6! + x8/8! - ...) + i(x - x3/3! + x5/5! - x7/7! + ...).
These two expressions are equivalent to the Taylor series of cos(x) and sin(x). So, plugging those functions into the expression gives cos(x) + isin(x).
Q.E.D.
Of course, were you to make x = п in Euler's formula, you'd get Euler's identity:
eiπ = -1
It depends on your definitionThe question presumes that someone has already given a definition of the exponential function for complex numbers. But has this definition been given? If so, what is this official definition?The answer above assumes that the exponential function is defined, for all complex numbers, by its power series. (Or, at least, that someone else has already proved that the power series definition is equivalent to whatever we're taking to be the official definition).
So the answer depends critically on what the definition of the exponential function for complex numbers is.
Suppose you know everything about the real numbers, and you're trying to build up a theory of complex numbers. Most people probably view it this way: Mathematical objects such as complex numbers are out there somewhere, and we have to find them and work out their properties. But mathematicians look at it slightly differently. If we can construct a thing which has all the properties we feel the field of complex numbers should have, then for all practical purposes the field of complex numbers exists. If we can define a function on that field which has all the properties we think exponentiation on the complex numbers should have, then that's as good as proving that complex numbers have exponentials. Mathematicians are comfortable with weird things like complex numbers, not because they have proved that they exist as such, but because they have proved that their existence is consistent with everything else. (Unless everything else is inconsistent, which would be a real pain but is very unlikely.)
So how do we construct this exponential function? One approach would be simply to define it by exp(x+iy) = ex(cos(y)+i.sin(y)). Then the answer to this question would be trivial: exp(iy) = exp(0+iy) = e0(cos(y)+i.sin(y)) = cos(y)+.sin(y). But you'd still have some work to do to prove things like exp(z+w) = exp(z).exp(w). Alternatively, you could define the exponential function by its power series. (There's a theorem that lets you calculate the radius of convergence, and that tells us the radius is infinite, i.e. the power series works everywhere.) Or maybe you could try something like proving that the equation dw/dz = w has a unique solution up to a multiplicative constant, and defining the exponential function to be the solution which satisfies w=1 at z=0.
Let z = cos(x) + i*sin(x)
dz/dx = -sin(x) + i*cos(x)
= i*(i*sin(x) + cos(x))
= i*(cos(x) + i*sin(x))
= i*(z)
therefore dz/dx = iz
(1/z)dz = i dx
INT((1/z)dz = INT(i dx)
ln(z) = ix+c
z = e(ix+c)
Substituting x=0, we get cos(0) + i*sin(0) = e(0i+c)
1+0 = e(0+c)
e(0+c) = 1
therefore c=0
z = eix
eix = cos(x) + i*sin(x)
Substituting x=pi, we get e(i*pi) = -1 + 0
e(i*pi) = -1
e(i*pi) + 1 = 0
qED
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No it is a "discrete" distribution because the outcomes can only be integers.
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The harmonic series is a sequence of notes and frequencies. These notes occurr naturally, even resonating in natural cave formations and cathedrals. It has to do with the natural fractions of a frequency. Imagine a string vibrating. It creates a kind of elongated "jump rope" shape. Now imagine that we have placed a weight in that jumprope so that it is held down perfectly in half causing each side to rotate separately but in unison. This devision causes the note to jump up an octave on a stringed instrument. When you continue to make these devisions in the jump rope, (halfs, thirds quarters fifths, etc) the notes will sound in a specific series, starting with the first note (or open string), then a perfect octave above that, then a perfect fifth above that, then a perfect fourth, Major Third, Minor Third etc.
The issue of the harmonic series is a complicated one because as one progresses up the natural order of fractions, the notes become imperfect or out of tune with one another. This is why today we have a tempered tuning system for our pianos instead of a harmonic series tuning system!
There u go
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Discrete
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It follows the rules of the home team.