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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 is the formula of population growth?
There is no simple formula for population growth.
Some of the factors that you need to take account of are:
Emigration rates
Immigration rates
Demographics
Age-specific fertility rates
Death rates
What is the infinite geometric series of 6?
Let r be any real number such that |r| < 1 and let a = 6 - 6r.
Then the geometric sequence: a, ar, ar^2, ar^3, ... will converge to 6.
Since the choice of r is arbitrary within the given range, there are infinitely many possible answers.
Differentiate between arithmetic series and geometric series?
In an arithmetic series, each term is defined by a fixed value added to the previous term. This fixed value (common difference) may be positive or negative.
In a geometric series, each term is defined as a fixed multiple of the previous term. This fixed value (common ratio) may be positive or negative.
The common difference or common ratio can, technically, be zero but they result in pointless series.
What is the Probability density function of Poisson distribution?
If a random variable X has a Poisson distribution with parameter l, then the probability that X takes the value x is
Pr(X = x) = lx*e-l/x! for x = 0, 1, 2, 3, ...
Why is it said that poisson distribution is a limiting case of binomial distribution?
This browser is totally bloody useless for mathematical display but...
The probability function of the binomial distribution is P(X = r) = (nCr)*p^r*(1-p)^(n-r) where nCr =n!/[r!(n-r)!]
Let n -> infinity while np = L, a constant, so that p = L/n
then
P(X = r) = lim as n -> infinity of n*(n-1)*...*(n-k+1)/r! * (L/n)^r * (1 - L/n)^(n-r)
= lim as n -> infinity of {n^r - O[(n)^(k-1)]}/r! * (L^r/n^r) * (1 - L/n)^(n-r)
= lim as n -> infinity of 1/r! * (L^r) * (1 - L/n)^(n-r) (cancelling out n^r and removing O(n)^(r-1) as being insignificantly smaller than the denominator, n^r)
= lim as n -> infinity of (L^r) / r! * (1 - L/n)^(n-r)
Now lim n -> infinity of (1 - L/n)^n = e^(-L)
and lim n -> infinity of (1 - L/n)^r = lim (1 - 0)^r = 1
lim as n -> infinity of (1 - L/n)^(n-r) = e^(-L)
So P(X = r) = L^r * e^(-L)/r! which is the probability function of the Poisson distribution with parameter L.
Is a divergent infinite series to the power of 2 also a divergent series?
Not necessarily, and I'll give you an example.
The harmonic series, Σ∞n=1 (1/n), is divergent.
However, if you square (1/n) and use the result in the above series; i.e. Σ∞n=1 (1/n2), which is the p-series for p = 2, the result is that the series converges, and so therefore, by definition, is not divergent.
How to Derive variance of Poisson distribution?
If X has the Poisson distribution with mean l
then Pr(X = k) = e-llk/k!
Mean of Poisson = Sum over all k of [k*P(X = k)] which happens to be l.
= Sum over all k of [k*e-llk/k!]
= Sum over all k of [e-llk/(k-1)!]
= Sum over all j of [le-llj/j!] where j has been substituted for k-1
= l*Sum over all j of [e-llj/j!]
But the quantity being summed is simply the pdf of the Poisson distribution and so its sum over all possible values is 1
So Mean = l
And then Variance of Poisson = Sum over all k of [k2*P(X = k)] - l2.
= Sum over all k of [k2*e-llk/k!] - l2
Then, since k2 = k*(k-1) + k
Variance = Sum over all k of [k*(k-1)e-llk/k!] +Sum over all k of [k*e-llk/k!] - l2
= Sum over all j of [l2e-llj/j!] where j has been substituted for k-2
+ Sum over all i of [le-lli/i!] where i has been substituted for k-1 - l2
= l2*Sum over all j of [e-llj/j!] + l*Sum over all i of [e-lli/i!] - l2
And since the sums are equal to 1,
Variance = l2 + l - l2 = l
Apologies: I did the answer using the symbol for lambda but this browser changed them all back to l. I cannot change them all t o something else but I hope it is clear. At least they are distinguishable from 1!
Are the mean and standard deviation equal in a poisson distribution?
The mean and variance are equal in the Poisson distribution. The mean and std deviation would be equal only for the case of mean = 1. See related link.
How is the pascal triangle and the binomial expansion related?
If the top row of Pascal's triangle is "1 1", then the nth row of Pascals triangle consists of the coefficients of x in the expansion of (1 + x)n.
Relationship between Exponential and Poisson Distributions?
Poisson distribution shows the probability of a given number of events occurring in a fixed interval of time. Example; if average of 5 cars are passing through in 1 minute. probability of 4 cars passing can be calculated by using Poisson distribution.
Exponential distribution shows the probability of waiting times between occurrences of events.
If we use the same example; probability of a car coming in next 40 seconds can be calculated by using exponential distribution.
-Poisson : probability of x times occurrence
-Exponential : probability of waiting times between events.
How is Pascal's triangle related to the Binomial Theorem?
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+x2 and look at the coefficients of the results you will see that they are 1, 2, 1. Now do it for (1+x)3 and they are 1, 3, 3, 1. These, of course, are the lines from Pascal's Triangle. I put the first part of the triangle below (in left-justified form). You should notice that when the exponent is 2, we use the third line and when the exponent is 3, we use the fourth line of the triangle.
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
1 9 36 84 126 126 84 36 9 1
In case you wonder about the first two rows. Look at (1+x)0 and
(1+x)1, their coefficients come from those two rows.
Now look at (a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4 which is more general and of course the coefficients come from the 5th line of the triangle.
So the answer to the question is that if we look at the binomial (a + b)n
The n+1 row of Pascal's triangle gives us the coefficients of the expanded form of the binomial. Seeing the examples first often makes this easier to see and understand. It is the n+1 row because the first row of the triangle is any binomial with an exponent of 0.
How do you find the first term of a geometric series?
The answer depends on what information you have been provided with.
What is the difference between poisson distribution and poisson process?
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 (having a poisson process) occurring in a certain period of time.
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.
