Binomial Definition and 668 Threads

In probability theory and statistics, the binomial distribution with parameters n and p is the discrete probability distribution of the number of successes in a sequence of n independent experiments, each asking a yes–no question, and each with its own Boolean-valued outcome: success (with probability p) or failure (with probability q = 1 − p). A single success/failure experiment is also called a Bernoulli trial or Bernoulli experiment, and a sequence of outcomes is called a Bernoulli process; for a single trial, i.e., n = 1, the binomial distribution is a Bernoulli distribution. The binomial distribution is the basis for the popular binomial test of statistical significance.
The binomial distribution is frequently used to model the number of successes in a sample of size n drawn with replacement from a population of size N. If the sampling is carried out without replacement, the draws are not independent and so the resulting distribution is a hypergeometric distribution, not a binomial one. However, for N much larger than n, the binomial distribution remains a good approximation, and is widely used.

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  1. N

    Binomial Expansion: Problem/Solution Explained

    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))...
  2. J

    Binomial theorem for fractional exponents?

    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!
  3. F

    Probability of X in Binomial Distribution B(20, 1/52)

    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...
  4. P

    Exploring the Binomial Coefficient Formula: n choose k | Homework Explanation

    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
  5. D

    Binomial Expansion (1+4x)^7: Guide for Correct Answer

    (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...
  6. R

    Binomial distribution notation

    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?)
  7. F

    Understanding the Binomial Coefficient through Generating Functions

    [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...
  8. G

    Expectation of Negative Binomial Distribution

    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...
  9. R

    Simplifying a trigonometric binomial

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

    How Accurately Can Binomial Series Approximate Sqrt(35)?

    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...
  11. S

    Efficiently Approximate sqrt(35) Using Binomial Series | 10^-7 Accuracy

    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 +...
  12. S

    Approximating Binomial Series I'm so close

    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...
  13. S

    Solve Binomial Theorem Problem: a+2b+4c=10^30

    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...
  14. T

    Binomial Theorem expansion with algebra

    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...
  15. D

    Proving the Binomial Theorem: Simplifying Binomial Coefficients

    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...
  16. S

    Binomial and geometric distributions

    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)...
  17. G

    Proof of Haldane's Binomial Expansion

    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...
  18. P

    How can I prove this binomial identity?

    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...
  19. S

    Binomial Expansion of (1+x)^n: Coefficient of x^n

    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...
  20. T

    Help: sum of binomial coefficents

    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...
  21. T

    Help: sum of binomial coefficents

    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...
  22. S

    Pascal's Triangle and Binomial Theorem

    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...
  23. MathematicalPhysicist

    Proving the Binomial Identity: A Shorter Solution Approach

    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...
  24. L

    Unbised estimator of Binomial Distribution

    [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...
  25. Simfish

    How is the negative binomial the inverse of the binomial distribution?

    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...
  26. N

    Advice on Exponential, Binomial, & Normal Distributions

    Hey, I'm new to all this so cut me some slack, but have been trying to work through some questions, and I can't seem to find answers to these questions... Or atleast find the confidence that my answers/working is correct... 1. (Exponential Distribution) Telephone calls arrive at the...
  27. C

    What is the Probability of Gerry Infield Getting Two Hits in His Next 5 At Bats?

    Homework Statement Gerry Infield has batting average of 0.326 what is the probability that he will have two hits in his next 5 times at bat?Give your answer correct to 3 decimal places. The Attempt at a Solution It is a binomial distribution with n trails and x sucesses (hence, n-x...
  28. E

    Verifying Binomial Equations Easily

    Homework Statement My book says that one "easily" verifies that (x+y)^n = (x + y)^(n-2)Q+(x+y)^(n-3)P where Q = x^2 + xy +y^2 and P = xy^2 + x^2y Homework Equations The Attempt at a Solution I began by expanding everything into summations with binomial...
  29. J

    Interval of convergence with binomial coefficient

    Homework Statement Find the radius of converge of: \sumx^{}(n choose k) Homework Equations Radius of converge = 1/limsup|an+1/an|, for power series: anx^n The Attempt at a Solution Tried rewriting (n choose k) as: n!/[k!(n-k)!] but where do I go from...
  30. C

    Caculate the probability using a binomial distribution

    Ok so I have a problem I am not sure of the method I should use. In a recent survey, 60% of the population disagreed with a given statement, 20% agreed and 20% were unsure. Find the probability of having at least 5 person who agree in a mini-survey with 10 people. I tried to caculate the...
  31. C

    Summation Proof with Binomial Theorem

    Prove the following statement: \[ \sum\limits_{r + s = t} {\left( { - 1} \right)^r \left( \begin{array}{c} n + r - 1 \\ r \\ \end{array} \right)} \left( \begin{array}{c} m \\ s \\ \end{array} \right) = \left( \begin{array}{c} m - n \\ t \\ \end{array} \right) \] Any initial...
  32. N

    Can the Binomial Theorem Prove This Series Equals (-2)^n?

    Use binomial theorem to prove C(n,0) - 3(C(n,1)) + 9(C(n,2) - 27(C(n,3) + ... + (-3)^n(C(n,n) = (-2)^n From looking at the data given b = (-3) so a = 1 so (-2)^n = (1-3)^n With this I know the equation in sigma notation and could probably prove the theorem through mathematical induction but...
  33. P

    Is the Binomial Theorem Applicable to All Real Numbers?

    I learned the Binomial Theorem a while ago. But it is only now that I think about how it is only useful for powers that are natural numbers. Can it be extended to all real numbers - e.g. 1/2?
  34. D

    Binomial Theorem and Electric Field question

    Homework Statement I'm having a lot of trouble solving for part b as I am unable to correctly apply the binomial theorem to this approximation. The problem is shown below: Three point charges are distributed: a positive charge +2Q in the center, and a pair of negative charges -Q, a...
  35. C

    How Do You Solve a Binomial Expansion Problem Involving Coefficients?

    Hi. I'm completely stuck on the following question, and have no idea how to even start it. Any help would be really appreciated. The first four terms, in ascending powers of x, of the binomial expansion of (1 + kx)^n are 1 + Ax + Bx^2 + Bx^3 + ..., where k is a positive constant...
  36. V

    DISCRETE MATH: Binomial Theorem proof (using Corollary 2)

    Homework Statement Show that if n is a positive integer, then 1\,=\,\binom{n}{0}\,<\,\binom{n}{1}\,<\,\cdots\,<\,\binom{n}{\lfloor\frac{n}{2}\rfloor}\,=\,\binom{n}{\lceil\frac{n}{2}\rceil}\,>\,\cdots\,>\binom{n}{n\,-\,1}\,>\,\,\binom{n}{n}\,=\,1 Homework Equations I think this proof involves...
  37. B

    Solving Binomial Expansions: Coefficient of x^k in (2x-1/x)^2007

    Hi, Im having some troubles with this binomial expansion... Determine the coefficient of x^k, where k is any integer, in the expansion of (2x - 1/x)^2007. I figured it would just be C(2007,k) * (2x)^2007-k * (-1/x)^k = C(2007,k) * (2)^2007-k * x^2007-k * (-1/x)^k therefore the...
  38. C

    Sum the Binomial Series: C_0^2-2C_1^2+...+(-1)^n(n+1)C_n^2

    Homework Statement Sum the following series to N terms: C_0^2-2C_1^2+...+(-1)^n(n+1)C_n^2 Arrgh! This is a very frustrating question. I have to multiply two series and find the coefficient of some term but I don't know what to do. Please don't ask me to give you some proof of my work at...
  39. B

    How Do You Expand (2c - 3d)^5?

    I'm really confused when it comes to this stuff. I'm stuck on this problem: (2c - 3d)^5 I have no idea where to begin. Please help!
  40. F

    Expand (1-6x)^4 (1+2x)^7 - Solve Binomial Expansion

    Homework Statement Expand (1-6x)^4 (1+2x)^7 in ascending powers of x up to and including the terms in x^3 Homework Equations (1-6x)^4 (1+2x)^7 The Attempt at a Solution Firstly, I expand (1-6x)^4 = (1)^4 + 4C1 (-6x) + 4C2 (-6x)^2 + 4C3 (-6x)^3 + ...
  41. W

    Binomial Theorem and Induction with Trigonometry

    Ok i need help to calculate the co-efficients of certain terms in the binomial expansion for example: (3 + (5/X)^2)^10 what is the coefficient of x^8? I hope that question works sorry if it doesn't i did just make it up then... if you know of any like it please help! also, an...
  42. S

    What is binomial distribution and how does it work?

    Hi, Can anyone explain binomial distribution to me. I tried wikipedia and some googling, but I just do not understand much of it. I don't come from maths background, I am more like an IT person. I need to write a short program calculating binomial distributions, however, first I need to...
  43. G

    Connection between cubed binomial and summation formula proof (for squares)

    I was reading through a proof of the summation formula for a sequence of consecutive squares (12 22 + 32 + ... + n2), and the beginning of the proof states that we should take the formula: (k+1)3 = k3 + 3k2 + 3k + 1 And take "k = 1,2,3,...,n-1, n" to get n formulas which can then be...
  44. M

    How to Simplify (11-1)^9 Using Binomial Expansion?

    Hello everyone. I'm studying for my exam and I'm reveiwing some problems but this one isn't making sense to me: 11^9(9 choose 0) - 11^8(9 choose 1) + 11^7(9 choose 2) - ... - 11^2(9 choose 7) + 11^1 (9 choose 8) - 11^0 (9 choose 9) answer: (11-1)^9 = 10^9 = 1,000,000,000. work...
  45. D

    Binomial Expansion Question Driving Me Mad

    Binomial Expansion Question Driving Me Mad! http://img489.imageshack.us/img489/5239/binomialiv6.jpg This is the last question on my maths sheet, and i must have been staring at it for hours, I've read all my notes and book but i just can't piece it together at all. Driving me crazy...
  46. A

    A quick question on coefficient of binomial expansion

    Is it always necessary that C(n,r) is an integer if n and r are integers ? Is there any proof ?Please clarify. thanks.
  47. N

    Suppose x is a discrete, binomial random variable

    How do I do this p(x<1) this sign has a _ under the < n=6 p=0.1 Suppose x is a discrete, binomial random variable. Calculate P(x > 2), given trails n = 8, success probability p = 0.3 [Hint: P(x > value) = 1 – P(x <= value) <= is a < with a _ under it (tell me the number...
  48. mattmns

    Combinatorics - Binomial Theorem Questions

    There are a few questions that have been giving me trouble with this binomial theorem stuff. (1). Using the binomial theorem and the relation (1+x)^{m_1} (1+x)^{m_2} = (1+x)^{m_1 + m_2} prove that: \sum_{k=0}^n \binom{m_1}{k} \binom{m_2}{n-k} = \binom{m_1 + m_2}{n} (2). Prove by induction...
  49. C

    Binomial Series Simplification

    How does one deduce the following: We are given \binom{-3}{n} = \frac{(-3)(-4)(-5)\cdot\cdot\cdot\cdot\cdot(-3-n+1)}{n!} = \frac{(-3)(-4)(-5)\cdot\cdot\cdot\cdot[-(n+2)]}{n!} . How do we get from here to: \frac{(-1)^{n}\cdot2\cdot3\cdot4\cdot5\cdot\cdot\cdot\cdot(n+1)(n+2)}{2\cdot n!} =...
  50. mattmns

    Integrating Binomial expansion

    Here is the question from the book: By integrating the binomial expansion, prove that, for a positive integer n, \frac{2^{n+1} - 1}{n+1} = 1 + \frac{1}{2}\binom{n}{1} + \frac{1}{3}\binom{n}{2} + ... + \frac{1}{n+1}\binom{n}{n} ------------ So I integrated both sides of the following...
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