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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
Does a sum of infinite ones equal a sum of infinite twos?
Yes, the sum of infinite ones equal the sum of infinite twos.
Derive the moment generating function of the poisson distribution?
The probability mass function (pmf, you should know this) of the Poisson distribution is
p(x)=((e-λ)*λx)/(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
(etx)*p(x), where t is a random variable
Therefore,
(etx)*((e-λ*λx)/(x!))
(e-λ)*sum[(e-λ*λx)/(x!)]
Thee-λ is only a constant; thus, it can be pulled out of the sums.
Continuing,
(e-λ)*sum[(λ*et)x)/x!]
Let y=λ*et
(e-λ)*sum[(y)x/x!]
By Macalurins series, the sum[(yx)/x! ]= ey
Soonwards
(ey)*(e-λ)
Lets return the y by λ*et
What is the sum of the infinite geometric series?
The sum of the series
a + ar + ar2 + ... is a/(1 - r) for |r| < 1
How can you tell if a infinite geometric series has a sum or not?
The geometric series is, itself, a sum of a geometric progression. The sum of an infinite geometric sequence exists if the common ratio has an absolute value which is less than 1, and not if it is 1 or greater.
How many experimental outcomes are possible for the binomial and the Poisson distributions?
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.
How do you find the coefficients of the terms in the binomial expansion?
The coefficient of x^r in the binomial expansion of (ax + b)^n isnCr * a^r * b^(n-r)
where nCr = n!/[r!*(n-r)!]
What is the difference between poisson and binomial distribution?
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 with a fixed time interval and having a fixed average rate then it is Poisson distribution.
2203 in base 10, converted to base 5 is 323032203 in base 5, converted to base 10 is 303.
What is the meaning of quantitative variables?
They are variables that can take quantitative - as opposed to qualitative values. For example, the colour of peoples' eyes is a qualitative variable, but their age or shoe size are quantitative variables.
What is the value of x in series 2 2 9 9 25 25 x?
Use the rule
U2n-1 = (9n2 - 13n + 8)/2
U2n = (9n2 - 13n + 8)/2
The next number is 50.
What does Geometric Series represent?
A geometric series represents the partial sums of a geometric sequence. The nth term in a geometric series with first term a and common ratio r is:T(n) = a(1 - r^n)/(1 - r)
This site no longer allows me to enter subscripts so I will use brackets: a(n) to indicate the nth term.
a(n) = a(1) + (n-1)*d where d is the common difference between the terms of the arithmetic sequence.
Therefore, d = [a(n) - a(1)]/(n-1)
Then, the appropriate arithmetic series is
S(n) = 1/2*n[2*a(1) + (n-1)*d] where all the terms on the right hand side are known.
1,424,007,025.076
Yes. A cubic equation can have 3 real roots. Depending on their size, each of three intervals could contain a root. In that case different intervals must give different roots.
Yes. A cubic equation can have 3 real roots. Depending on their size, each of three intervals could contain a root. In that case different intervals must give different roots.
Yes. A cubic equation can have 3 real roots. Depending on their size, each of three intervals could contain a root. In that case different intervals must give different roots.
Yes. A cubic equation can have 3 real roots. Depending on their size, each of three intervals could contain a root. In that case different intervals must give different roots.
How do you find the sum of a geometric series?
Let's say Un=aqn and Sn=a+aq+aq2+aq3+aq4+aq5+...+aqn
Sn = a (1+q+q2+q3+q4+q5+...+qn)
A=(Sn/a) - q (Sn/a) = (1+q+q2+q3+q4+q5+...+qn) - q(1+q+q2+q3+q4+q5+...+qn)
A=1+q+q2+q3+q4+q5+...+qn-q-q2-q3-....-qn-qn+1=1-qn+1
So A = 1-qn+1 = Sn/a (1-q)
So Sn = a (1-qn+1)/(1-q)
What is the relationship between the binomial expansion and binomial distribution?
First i will explain the binomial expansion
Interval estimation of the parameters of poisson distribution?
n order to fit the Poisson distribution, we must estimate a value for λ from the observed data. Since the average count in a 10-second interval was 8.392, we take this as an estimate of λ (recall that the E(X) = λ) and denote it by ˆλ.
That is the correct spelling of "fourths" (meaning the plural of one-fourth).
What does culture convergence mean?
Cultural convergence is when the similarities between different cultures in the same nation are driven by global and common brands. The similarities are with aspects like consumer preferences, beliefs, aspirations, and values.
