Binomial Definition and 640 Threads

Rudin's proof of lim n-> inf (p^(1/n)) = 1 1+n*x_n <= (1 + x_n)^n = o I don't see it from the binomial theorem, which is what he says that is from. He also does things with the binomial theorem like: (1+x_n)^n >= ((n(n-1)) / 2) *x_n^2 I'm not sure what he did to get these two...

I've started listening to the lectures for the MIT OpenCourseWare 18.01 Single Variable Calculus class. I understood all of it up until the teacher found the derivative of xn. Here's what he wrote on the board: \frac{d}{dx} x^{n} = \frac{\Delta f}{\Delta x} = \frac{(x+\Delta x)^{n} -...

Hey people, I've racked my brain on this question for hours and can't seem to get to grips with it, wondering if i could get a little guidance? Homework Statement Considering the co-efficient of x^n in the expansion of (1+x)^n(1+x)^n, show: [the sum from k=0 -> k=n of:] [nCk]2 = 2nCn...

Hey guys, Can anyone please explain the differences between binomial and poisson distribution. THANK U>>>>>>>>>>>>>>>>>>>>>

is it possible to integrate the binomial theorem??

Homework Statement Describe a binomial experiment that can be solved using the expression 10C5 (0.2)5 (0.8)5 Homework Equations The Attempt at a Solution Have no idea.

Homework Statement Find the binary decimal expansion of the fraction 1/3. Identify the repeating decimals of digits. The Attempt at a Solution I have that 1/3=0.0101111... and so the repeating digit is 1. Is this right? It's the first time I've been exposed to binary expansion.

Homework Statement I am trying to figure out if I have a 20% chance to get what I want (let it = x) and I have 6 chances to do so (n=6), I am curious how I set this question up to find out my chances of getting 'x' once out of the 6 times I try. Homework Equations Binomial Distribution...

I know how to expand binomials with the aid of pascals triangle and also with the aid of the nCr function on the calculator. I'm not quite sure about this formula though see the part in the brackets where n is above k. What does that mean? Someone told me that represents nCk. Is that true...

I stuck at the second "="...i know it goes like this because the formula...i just someone explain to me why it works like that. thank you soooo much!

Homework Statement a) Expand ([2/3]+[1/3])^4 b) Four chocolates are randomly taken (with replacement) from a box containing strawberry creams and almond centres in the ratio 2 : 1. What is the probability of getting: i) all strawberry creams ii) two of each type iii) at least 2...

Use the Binomial Theorem to show that n summation k-0 3^k C(n,k) = 2^2n hint of the question is 3^k C(n,k) = 3^k 1^n-k C(n,k)

how do i use binomial to show that 3^k C(n,k) = 3^k 1^n-k C(n,k)

Ok, so after a little discussion with my discrete math teacher today, he sent me on a little "quest". Here is how it happened: The topic we were covering was set theory, and as I had been studying very basic combinatorics the night before, I noticed something about the powerset, namely...

Use the binomial expansion of (1+x)^(-1/2) to find an approximation for 1/(rt4.2). I've got the expansion of (1+x)^(-1/2) as 1-(1/2)x+(3/8)x^2... but the obvious idea of substituting x=3.2 gives me the wrong answer. I think it's something to do with the expansion being valid but can't...

Ok, the question says Binomial with n=40. p=0.6 use normal approximation and determine.. a) value at 14. b)value lest then 12. So I thought I had this down and packed but the answer in the back of the book tells me otherwise. Anyways the following is my working. Y~Nor(13.5<x<14.5)...

Homework Statement A population has an average of 12 defects per 100 feet of wire sampled and inspected. What is the probability of finding 20 or fewer defects in a sample? Homework Equations I think I am supposed to use the binomial distribution b(x;n,p) The Attempt at a...

Hi everyone, I have been having a problem with the General Binomial Coefficient for any rational value: \left( \begin{array}{c} n\\ r\end{array} \right) = \frac{1}{r!}\prod_{i=0}^{r-1} (r-i) Now this works fine except when r=0. so 0! is defined to be 1 so the coefficient of the...

kx = x !/r! (x – r)! kx− r is that right?

Binomial, Poisson and Normal Probability distribution help! Hey everyone, I've just started a new section on probability (argh) in my math course, and unlike most other maths, I cannot cope! Anyway, I was wondering if anyone could tell me if I did the following question correctly! Any helps...

Homework Statement If \sum^{n}_{r=0} \frac{1}{^{n}C_{r}} = a, then find the value of \sum^{n}_{r=0} \frac{r}{^{n}C_{r}} in terms of a and n.[/tex] The Attempt at a Solution I tried to write down the terms of both the series, but to no avail. i can't think of...

Homework Statement \sum^{n}_{r=0} (2r+1) (^{n} C_{r})^{2} The Attempt at a Solution x(1+x^{2})^{n} If I differentiate this and put x=1; I will get the above series without the squares of the binomial coefficients.Will multiplying by (1+x)^{n} help now?

Homework Statement Evaluate \sum^{m}_{r=0} ^{ n + r }C_{n} I can handle things when the lower thing in the combination part is changing, what shall I do with this one?

Homework Statement If p is nearly equal to q and n>1, show that \frac{(n+1)p+(n-1)q}{(n-1)p+(n+1)q}=(\frac{p}{q})^{\frac{1}{n}} Note: the index 1/n is on the whole fraction (p/q) I think it might be helpful if I specify th chapter from which I got this question. Its the binomial thorem...

Question: Find the coefficient of x^5 in (1+x+x^2)^4. Problem: I have not come across expanding brackets which have x^2. I know how to apply the binomial theorem for (a+b)^n or (1+a)^n but have not come across (1 + ax + ax^2)^n. They are not explained in my textbooks so I was wondering if...

http://img529.imageshack.us/img529/3044/bin2od4.th.jpg http://img529.imageshack.us/img529/9734/bin1ml5.th.jpg The problem and solution are attached above. Firstly, why is the expansion valid for (mod x/(1+x)) < 1, and not just for mod(x)<1? Also, I do not understand how from (mod x/(1+x))...

I am curious, is there any way to use the binomial theorem for fractional exponents? Is there any other way to expand a binomial with a fractional exponent? I suppose Newton's theorem is not a way since it requires factorials. Thanks!

http://img262.imageshack.us/img262/4669/rangevu0.jpg Ok so I think its asking for a binomial distribution of B(20, 1/52) Would this be the probability function of X? Also the range space of X is asked in a lot of these questions, but I've no idea on how to calculate it. I thought it...

Homework Statement How is possible this equality: {n \choose k} = \frac{n \cdot (n-1) \cdots (n-k+1)}{k(k-1)...1} = \frac{n!}{k!(n-k)!} ? I mean where the second part \frac{n!}{k!(n-k)!} comes from? Homework Equations The Attempt at a Solution

(1+4x)^7 i need to fully expand. il use (4c0) for 4 Ncr 0. (4c0)1^4 + (4c1) 1^3 + 4x + (4c2) 1^2 4x^2 + (4c3) 1 + 4x^3 + (4c4) 4x^4. this gives the completely wrong answer. i know this isn't the way your meant to do it, but i can't remember what's wrong here. can...

If the probability of a successful outcome is p and failure is q and there are n trials P(X=x)= ^n C_x p^xq^{n-x} and you write that as X~Bin(n,p) <--- How would I read that? (like what does the ~ mean?)

[SOLVED] Binomial Coefficient Homework Statement I'm reading the textbook http://www.math.upenn.edu/~wilf/DownldGF.html" , and in section 1.5, the author explains how to derive the binomial coefficient via a generating function approach. One of the things he mentions as obvious, isn't (at...

I am re-writing up some lecture notes and one of the proofs that E[X] for the negative binomial is r/p where r is the number of trials...The problem is there are a number of books that say r(1-p)/p is the correct expectation whilst others agree with 1/p Which one is correct...for what its...

Is this correct? (bcos30-85.7)(bcos30-85.7)=b²cos²30-85.7cos30-85.7bcos30+85.7²

Use binomial series to approximate sqrt (35) with an accuracy of 10^(-7) (35) = sqrt(35*36/36) = 6*sqrt(35/36) Formula: (1+x)^n where x=(-1/36) and n=(1/2): 6*sqrt(35/36) = 6[(1 + (- 1/36))^(1/2)] = The coefficients of the binomial series are: '1/2 choose 0' is 1. '1/2 choose 1' is...

Use binomial series to approximate sqrt (35) with an accuracy of 10^(-7) (35) = sqrt(35*36/36) = 6*sqrt(35/36) Formula: (1+x)^n where x=(-1/36) and n=(1/2): 6*sqrt(35/36) = 6[(1 + (- 1/36))^(1/2)] = from k=0 to k=4: = 6[(1 - (1/2)*(1/36) + (1/8)*(1/36)^2 - (1/16)*(1/36)^3 +...

Use Binomial Series to approximate sqrt(35) with an accuracy of 10^(-7) Formulas for binomial series: (1-x)^r and sum{from 0 to n}(r k)(x)^k sqrt(35) = sqrt(35*36/36) = 6*sqrt(35/36) = 6*sqrt(1-(1/36)) Now it looks more like the binomial series formula: let r = 1/2 because of the...

Homework Statement I am trying to find the number of nonnegative integer solutions to a+2b+4c=10^30. I found a generating function, and need to check the coefficient of 10^30. Homework Equations The generating function is 1/((1-x)(1-x^2)(1-x^4)). I found the PFD, which is...

In the binomial expansion (2k+x)^{n}, where k is a constant and n is positive integer, the coefficient of x² is equal to the coefficient of x³ a) Prove that n = 6k + 2 b) Given also that k = .\frac{2}{3}, expand (2k+x)^{n} in ascending powers of x up to and including the term in x³, giving each...

Homework Statement Show that binomial coefficients \frac{-1}{n} = (-1)^{n} Homework Equations (1+x)^p = (p / n) x^n The Attempt at a Solution I'm clueless on the idea of binomial coefficients. I think if I understood the question better I'd know at least where to start...

i was doing some exercises nut I'm not sure if my answers are correct 1) X~B(5,0.25) i have to find: a) E(x^2) and my answer was 2.5, is this correct? b) P(x(>or=to)4) and my answer was 0.0889, is this correct? 2) X~Geom(1/3) i have to find: a) E(x) my answer is 1/3 b) E(x^2) c)...

The next-to-last step in the proof on pg 1 of this article http://links.jstor.org/sici?sici=0006-3444%28194511%2933%3A3%3C222%3AOAMOEF%3E2.0.CO%3B2-Whttp://links.jstor.org/sici?sici=0006-3444%28194511%2933%3A3%3C222%3AOAMOEF%3E2.0.CO%3B2-W makes this substitution \sum_{r=0}^\infty...

Homework Statement Prove that the following binomial identity holds: {n+k-1 \choose k} = \sum_{i=1}^k {k-1\choose i-1}{n\choose i} The Attempt at a Solution One of the methods I've tried is to use induction on the variable n, but while trying this I got stuk on rewriting the...

Apply the binomial expansion to : (1+x)^n and show that the coefiicient of x^n in the expansion of (1+x)^2n is: (nC0)^2 +(nC1)^2 +...+(nCn)^2 hint: (nCm)=(nC(n-m)) my approach: (1+x)^n = x^n + nx^(n-1) + (nC2)x^(n-2) +...+ 1 (1+x)^2n = x^(2n) + nx^(2n-1) +...+ x^n i don't know what...

Help: sum of binomial coefficents ! Hello! I cannot figure out how to derive a closed formula for the sum of "the first s" binomial coefficients: \sum_{k=0}^{s} \left({{n}\atop{k}}\right) with s<n Could you please help me find out some trick to derive the formula... I've an exam on...

Help: sum of binomial coefficents ! Hello! I cannot figure out how to derive a closed formula for the sum of "the first s" binomial coefficients: \sum_{k=0}^{s} \left({{n}\atop{k}}\right) with s<n Could you please help me find out some trick to derive the formula... I've an exam on...

1. Evaluate the numbers for the coefficient of x4y9 in the expansion of (3x + y)13. 2. The Binomial Theorem states that for every positive integer n, (x + y)n = C(n,0)xn + C(n,1)xn-1y + ... + C(n,n-1)xyn-1 + C(n,n)yn. 3. I understand that the coefficients can be found from the n row of...

problem prove that: \forall n \in N \forall 0<=k<=2^{n-1} (C(2^n,k)=\sum_{j=0}^{k}C(2^{n-1},j)C(2^{n-1},k-j)) attempt at solution induction seems to be too long I am opting for a shorter solution, so the sum that it's wrriten in the rhs is the square of the sum of the term C(2^(n-1),j) but...

[SOLVED] unbised estimator of Binomial Distribution I have no idea how to find such an estimator SupposeX_1, ..., X_n \sim Bern(p) find an unbiased estimator of p^m, for m < n Induction on m was a nasty mess that should not be expected. The power of m causes some problem when I try to go...

Can anyone give a user-friendly explanation? http://en.wikipedia.org/wiki/Negative_binomial_distribution#Properties We see that the binomial distribution measures the probability of X successes after n trials, whereas the negative binomial measures the probability of the trial number after...