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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
How can the Poisson distribution be applied to a continuous frequency distribution?
Let L(t) be the instantaneous average rate of occurrences per unit time, at time t. So, for the ordinary Poisson distribution with parameter L, we just have L(t)=L for all t.
Let I be the integral of L(t) dt over a certain time interval [0,T], say.
Then, assuming that L(t) is continuous, or maybe just Riemann integrable, the total number of occurrences during [0,T] simply follows a Poisson distribution with parameter I. This is the simple answer one might expect.
To prove this (SKETCH: further estimates are needed to make this really rigorous): divide [0,T] into many small intervals [tj, tj+1). In each interval, the number of occurrences is approximately Poisson with parameter L(tj)(tj+1-tj).
The occurrences in each small interval are all independent of each other; hence the total number in [0,T], which is the sum of all these, follows a Poisson distribution with parameter the sum of L(tj)(tj+1-tj).
As you make the maximum size of the intervals shrink to zero, this sum tends towards I, the Riemann integral of L(t)dt over [0,T], as required.
What is the assembly program to generate a geometric series and compute its sum The inputs are the base root and the length of the series The outputs are the series elements and their sum?
Moment generating and the cumulant generating function of poisson distribution?
The moment generating function is
M(t) = Expected value of e^(xt)
= SUM[e^(xt)f(x)]
and for the Poisson distribution with mean a
inf
= SUM[e^(xt).a^x.e^(-a)/x!]
x=0
inf
= e^(-a).SUM[(ae^t)^x/x!]
x=0
= e^(-a).e^(ae^t)
= e^[a(e^t -1)]
Who was the first person to study determinants and discovered Bernoulli's numbers before Bernoulli?
T.seki
Why is this distribution referred to as a geometric distribution?
The geometric distribution is:
Pr(X=k) = (1-p)k-1p for k = 1, 2 , 3 ...
A geometric series is a+ ar+ ar2, ... or ar+ ar2, ...
Now the sum of all probability values of k = Pr(X=1) + Pr(X = 2) + Pr(X = 3) ...
= p + p2+p3 ... is a geometric series with a = 1 and the value 1 subtracted from the series.
See related links.
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
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
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?
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 are the properties of poisson distribution?
It is a discrete distribution in which the men and variance have the same value.
Find root ninety eight using Taylor's series?
The first-order Taylor expansion around x2 is given by sqrt(x2 + a) ~ x + a / 2x
So for sqrt(98), we know the closes perfect square is 81 = 9^2. Therefore: sqrt(98) = sqrt(81 + 17) = sqrt(9^2 + 17) ~ 9 + 17/(2*9) = 9 + 17/18 ~ 9.9444
Alternatively, sqrt(98) = sqrt(100 - 2) = sqrt(10^2 - 2) ~ 10 - 2/(2*10) = 10 - 1/10 = 9.9
Using a calculator: sqrt(98) ~ 9.899
Taylor series will always be an over-approximation.
