Parent Category: Numerical Analysis and Simulation

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.

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...

The geometric distribution is: Pr(X=k) = (1-p) k-1 p for k = 1, 2 , 3 ... A geometric series is a+ ar+ ar 2 , ... or ar+ ar 2 , ... Now the sum of all probability values of k = Pr(X=1) + Pr(X = 2) + Pr(X = 3) ... = p + p 2 +p 3 ... is a geometric series with a = 1 and the value 1...

Error propagation in numerical analysis is just calculating the uncertainty or error of an approximation against the actual value it is trying to approximate. This error is usually shown as either an absolute error, which shows how far away the approximation is as a number value, or as a relative...

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...

When things expand, it can block certain things out like a machine

\n \n Normal \n 0 \n 21 \n \n \n false \n false \n false \n \n \n \n \n \n \n \n MicrosoftInternetExplorer4 \n \n \n\n The fundamental =\n1st harmonic is not an overtone! \n\n \n\n Fundamental\nfrequency = 1st harmonic. \n\n 2nd harmonic = 1st\novertone. \n\n 3rd harmonic = 2nd...

Is the binomial expansion.

A poisson process is a non-deterministic process where events occur continuously and independently of each other. An example of a poisson process is the radioactive decay of radionuclides. A poisson distribution is a discrete probability distribution that represents the probability of events ...

The sum of the series a + ar + ar 2 + ... is a/(1 - r) for |r| < 1

It is quite complicated, and starts before Fourier. Trigonometricseries arose in problems connected with astronomy in the 1750s, andwere tackled by Euler and others. In a different context, theyarose in connection with a vibrating string (e.g. a violin string)and solutions of the wave equation. ...

A harmonic frequency is a multiple of the fundamental. If the frequency is 60Hz, then the 2nd harmonic is (2 * 60) 120 Hz, the third is (3 * 60) 180 Hz, etc.

The probability mass function (pmf, you should know this) of the Poisson distribution is .
p(x)=((e -Î» )*Î» x )/(x!), where x= 0, 1, ........ Then you take the expected value of exp(tx), you should always keep in mind to find the moment generating function (mgf) you must always do (e tx...

Application in String theory in Quantum Mechanics

Convergence of telecommunications

"Convergence in probability" is a technical term in relation to a series of random variables. Not clear whether this was your question though, I suggest providing more context.

Poisson distribution shows the probability of a given number ofevents occurring in a fixed interval of time. Example; if averageof 5 cars are passing through in 1 minute. probability of 4 carspassing can be calculated by using Poisson distribution. Exponential distribution shows the probability of...

PROGRAM :- /* Runge Kutta for a set of first order differential equations */ #include #include #define N 2 /* number of first order equations */ #define dist 0.1 /* stepsize in t*/ #define MAX 30.0 /* max for t */ FILE *output; /* internal filename */ void runge4(double x,...

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)]

No it is a "discrete" distribution because the outcomes can only be integers.

the Taylor series of sinx

Binomial Theorum

Answer .
Harmonics are multiples (thirds, fifths, etc) or divisions of frequencies. In radio, harmonics can be used carry additional signals on a single base frequency. It is the harmonics of an audio frequency that make a musical instrument unique. By damping a string at a half or third/fifth of...

769 853

If X and Y are i.i.d Poisson variables with lambda1 and lambda2 then, P (X = x | X + Y = n) ~ Bin(n, p) where p = lambda1 / lambda1 + lambda2

Probably the movement on a swing can be approximated by assuming that the magnitude of each swing will be a certain percentage of the previous swing (because of lost energy).

Simply because the Maclaurin series is defined to be a Taylor series where a = 0.

Rapidity of convergence is a relative concept whose meaning comes from the "comparison test" for convergence. One version of the comparison test says that if (an) is convergent and abs(bn)0, then (bn) must be convergent too.

You must pay for the answer

f(x)=lnx

Yes. If the Maclaurin expansion of a function locally converges to the function, then you know the function is smooth. In addition, if the residual of the Maclaurin expansion converges to 0, the function is analytic.

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.

L is 50 in roman numerals.

6. All even powers of 4 end in 6

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.)

Euler's formula states that e i x = cos(x) + i sin(x) where i is the imaginary number and x is any real number. First, we get the power series of e i x using the formula: e z = Î£ â n=o z n /n! where z = i x. That gives us: 1 + i x + ( i x) 2 /2! + ( i x...

As it has been already hinted, Fourier Series is used for periodic signals. It represents the signal by the discrete-time sequence of basis functions with finite and concrete amplitude and phase shift. The basis functions, according to the theory, are harmonics with the frequencies, divisible by the...

I think an example will help most people see it better than just an explanation/answer. So first a few examples are presented and than a general answer. Start with (1+x) 2 = 1+2x+x 2 and look at the coefficients of the results you will see that they are 1, 2, 1. Now do it for (1+x)...

Use the formula (n 2 - n)/2 where n is the given number of lines. That gives: (1 2 - 1)/2 = 0 for one line, (2 2 - 2)/2 = 1 for two lines, (3 2 - 3)/2 = 3 for three lines, (4 2 - 4)/2 = 6 for four lines, (5 2 - 5)/2 = 10 for five lines, and so on.

Given any number it is easy to find a rule based on a polynomial of order 5+k such that the first five numbers are as listed in the question and the next k are the given "next" numbers. However, use the rule U n = (-6n 4 + 83n 3 - 411n 2 + 874n - 468)/6 Accordingly, the next three numbers...

Boiling of materials like egg.

Not always true. Eg the divergent series 1,0,2,0,3,0,4,... has both convergent and divergent sub-sequences.

2.75 million = 2,750,000 or as 2.75*10 6 in scientific notation

4.95 = 4.95

The numbers are increasing in line with the 3 times (multiplication) table. 6-7 is plus 1. 7-10 is plus 3. 10-16 is plus 6. 16-25 is plus 9. Next in the 3 times table is 12. So, the next number will be 25 + 12. The answer is 37 .

Eleven million, to the nearest hundredths.

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).

noun 1. Finance. .
one part or division of a larger unit, as of an asset pool orinvestment: The loan will be repaidin three tranches..
a group of securities that share a certain characteristic andform part of a larger offering: The second tranche of thebond issue has a five-year maturity..
2. any...

Calculation a) Standard Deviation (s) is calculated as follows: .
ââ( X - Xi) 2 n-1 For bilateral tolerances: i) Capability Index (Cp) = USL - LSL / 6s ii) Performance Index or Centering of Process (CmK) = Minimum of USL- X or X - LSL 3s 3s For single sided...

It points to three or 3

The frequency is unrelated to the fall, so it may have any frequency. Or no frequency at all.

its got tod od somthing about maths that's all i know

Is it for any particular theme? Manipulating is one. Anti-technology, generally anything with 'ology' as a suffix.

Let's say U n =aq n and S n =a+aq+aq 2 +aq 3 +aq 4 +aq 5 +...+aq n S n = a (1+q+q 2 +q 3 +q 4 +q 5 +...+q n ) A=(Sn/a) - q (Sn/a) = (1+q+q 2 +q 3 +q 4 +q 5 +...+q n ) - q(1+q+q 2 +q 3 +q 4 +q 5 +...+q n ) A=1+q+q 2 +q 3 +q 4 +q 5 +...+q n -q-q 2 -q 3 -....-q n -q n+1 =1-q n+1 So...

You cannot because it does not exist. Although all the moments of the lognormal distribution do exist, the distribution is not uniquely determined by its moments. One of the consequences of this is that the expected values E [e^tX] does not converge for any positive t.

The binomial distribution is a discrete probability distribution. The number of possible outcomes depends on the number of possible successes in a given trial. For the Poisson distribution there are Infinitely many.

We often come across the algebraic identity (a + b) 2 = a 2 + 2ab + b 2 . In expansions of smaller powers of a binomial expressions, it may be easy to actually calculate by working out the actual product. But with higher powers the work becomes very cumbersome. The binomial expansion theorem is a...

A single number cannot define a series.

it is what we now call factor rationalising .

10001 and 9999, possibly.

Five hundred forty-three billion, two hundred one million, sixty-five thousand, four hundred eighty-three.

Using the Taylor series expansion of the exponential function. See related links

The value of 9 is 798,000 is in the ten-thousands place, so its value is 90,000.

The coefficient of x^r in the binomial expansion of (ax + b)^n is.
nCr * a^r * b^(n-r).
where nCr = n!/[r!*(n-r)!].

Poisson and Binomial both the distribution are used for defining discrete events.You can tell that Poisson distribution is a subset of Binomial distribution. Binomial is the most preliminary distribution to encounter probability and statistical problems. On the other hand when any event occurs...

2203 in base 10, converted to base 5 is 32303.
2203 in base 5, converted to base 10 is 303..