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Numerical Series Expansion
Mathematicians, scientists, and engineers often need solutions for difficult to unsolvable equations in order to progress in their work. Fortunately, through various methods of numerical and polynomial series expansions, some of the most problematic equations can be approximated to easily workable forms. Please post all questions regarding the various expansions and series, such as the Taylor and Maclaurin series expansions, the binomial expansion, and the geometric expansion, as well as the rules that govern them, into this category.
274 Questions
What are the unit labels for volume?
In the SI system, you'll find . . .
liter
cubic meter
milliliter
cubic centimeter.
But the 'customary' system is where you'll find
cubic inch
cubic foot
cubic yard
fluid ounce
cup
pint
quart
gallon
Why belong exponential family for poisson distribution or geometric distribution?
Why belong exponential family for poisson distribution
What is the Assumptions for a Binomial distribution and Poisson?
For the binomial, it is independent trials and a constant probability of success in each trial.
For the Poisson, it is that the probability of an event occurring in an interval (time or space) being constant and independent.
What does an equal in geometric series?
Geometric series may be defined in terms of the common ratio, r, and either the zeroth term, a(0), or the first term, a(1).
Accordingly,
a(n) = a(0) * r^n or
a(n) = a(1) * r^(n-1)
What is the comparison between pascal triangle expansion and binominal expansion?
Pascal's triangle shows the constant for each term if the equation is (x+y) to a number, which is the line number in Pascal's triangle, for a binomial expansion you can use Pascal's triangle but you have to multiply that by the constants on x and y raised to x's and y's exponent multiplied by the number the binomial is being raised to. (ax^b + cy^d) ^e = the number in Pascal's triangle for e times (a^ (b times e)) times (c ^(d times e)) which gives the constant for that term
What is Binomial Expansion and how does it relate to Pascal's Triangle?
The binomial expansion is the expanded form of the algebraic expression of the form (a + b)^n.
There are slightly different versions of Pascal's triangle, but assuming the first row is "1 1", then for positive integer values of n, the expansion of (a+b)^n uses the nth row of Pascals triangle. If the terms in the nth row are p1, p2, p3, ... p(n+1) then the binomial expansion is
p1*a^n + p2*a^(n-1)*b + p3*a^(n-2)*b^2 + ... + pr*a^(n+1-r)*b^(r-1) + ... + pn*a*b^(n-1) + p(n+1)*b^n
In basic mathematics, n factorial is equal to 1*2*3*...*n and is written as n! for positive integer values of n.
The Gamma function is a generalisation of this concept, with
Gamma(x) = (x-1)! where x can be any real or complex.
Find the first 5 derivatives of cos x1 x Use Maclaurin series to 6 terms?
Cos (aX) = 1 + bX, give value of X in terms of a and b.Practical Problem at site: An arc shape insert- plate of which R was 1250 and D was 625mm and thus S = 2618 (chord length (L)= 2165), has been damaged and flattered.
And site staff gives feedback that it is now D = 525 instead of 625 and Chord length =2280.but this all not match in a geometrical figure. One dimension is wrong. If we consider D= 525 is correct then what is new radius. S=2618 will remain unchanged. Hence if S and D is given what is new R.
By trigonometry and geometrically this relation will arrive like this
S = R . Theta ---------- Equation no. 1
where R is radius of arc and Theta is angle in radians of arc ends at centre and S is length of arc.
second relation if distance of arc centre to chord centre is = D, then
Cos (Theta/2) = (R - D)/R -------------Equation no 2
Simplifying both relation,
Cos (S/2R) = (R-D)/R
Here equation is with one unknown, because S and D is known variables. Only R is to find out.
In Simple form it can be written to solve further this equation is
Cos (aX) = 1 + bX, give value of X in terms of a and b. Please help to solve this simple Equation
R
L/2
D
Theta
Whats the mathematical basis of the harmonic series?
If a, b, c, d.......are in Arithmetic Progression (A.P.), then 1/a. 1/b, 1/c, 1/d.....are in Harmonic Progression (H.P.)
Who discovered pascal's triangle and binomial expansion before pascal?
Pascal's triangle appeared in some work by the Indian mathematician Pingala in the 2nd Century BC. Although details of Pingala's work are lost, the idea was subsequently expanded upon by Halayudha in the tenth Century. At around the same time, it was discussed by the Persian mathematician, Al-Karaji. It was also known to the Chinese mathematician Jia Xian in the eleventh Century. It is quite possible that the underlying combinatorial mathematics was known to earlier mathematicians but in any case, it is abundantly clear that Pascal was too late by over 3.5 Centuries. It says something about the Eurocentric writers that it is called Pascal's triangle, doesn't it?
It is expressing a number in decimal form: that is, a form in which the place value of each digit is one tenth the place value of the digit to its left.
How do you obtain the moment generating function of a Poisson distribution?
Using the Taylor series expansion of the exponential function.
See related links
What are the properties of poisson distribution?
It is a discrete distribution in which the men and variance have the same value.
let binomial be (a + b)now (a+b)3 will be (a+b)(a+b)2
= (a+b)(a2 + 2ab+ b2)
= a(a2+ 2ab+ b2) + b(a2 + 2ab+ b2)
= a3+ 2a2b+ ab2 + a2b + 2ab2 + b3
= a3+ 2a2b+ ab2 + a2b + 2ab2 + b3
= a3 +3a2b + 3ab2 +b3
hope it helped... :D
What is a binomial distribution?
The binomial distribution is one in which you have repeated trials of an experiment in which the outcomes of the experiment are independent, the probability of the outcome is constant.
If there are n trials and the probability of "success" in each trail is p, then the probability of exactly r successes is (nCr)*p^r*(1-p)^(n-r) :
where nCr = n!/[r!*(n-r)!]
and n! = n*(n-1)*...*3*2*1
A maclaurin series is an expansion of a function, into a summation of different powers of the variable, for example x is the variable in ex. The maclaurin series would give the exact answer to the function if the series was infinite but it is just an approximation.
Examples can be found on the site linked below.
What are the Situations wherein poisson distribution is applied?
The Poisson distribution may be used when studying the number of events that occur in a given interval of time (or space). These events must occur at a constant rate, be independent of the time since the previous occurrence.
What number is larger than the trillions?
now i MAY be wrong, but ill go ahead:
Million
Billion
Trillion
Quadrillion
Pentillin
Sectillion
(Don't know the one for 7)
Octillion
And that's all I can figure, hope I was help XP
Edit: It goes by the common scientific naming of multiplicity, just like what is described above, but you don't use the prefixes used for geometry like what is described. You instead use the common prefixes used in naming chemical compounds. Here's the list:
Million
Billion
Trillion
Quadrillion
Quintillion
Sextillion
Septillion
Octillion
Nonillion
Decillion
Undecillion
Duodecillion
and so on and so forth...duodecillion is actually 10,000,000,000,000,000,000,000,000,000,000,000,000,000 so if you need to really count anything more than that you're probably at a molecular level and would simply call it 10 to the 39th power.
:)
The gamma function is an extension of the concept of a factorial. For positive integers n, Gamma(n) = (n - 1)!The function is defined for all complex numbers z for which the real part of z is positive, and it is the integral, from 0 to infinity of [x^(z-1) * e^(-x) with respect to x.
If the outcomes of a random variable follow a Poisson distribution then their?
means equal the standard deviation
State a condition under which the binomial distribution can be approximated by poisson distribution?
Because "n" is very large and "p" is very small. where "n'' indicates the fixed number of item. And ''p'' indicates the fixed number of probability from trial to trial.
