What is Binomial: Definition and 667 Discussions

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. MarkFL

    MHB Abigail's question at Yahoo Answers regarding binomial expansion

    Here is the original question: Here is a link to the original question: What is the third term of the expansion of (2x+y^2)^9? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  2. W

    Expected Value and Binomial Random Variable

    1. In scanning electron microscopy photography, a specimen is placed in a vacuum chamber and scanned by an electron beam. Secondary electrons emitted from the specimen are collected by a detector and an image is displayed on a cathode-ray tube. This image is photographed. In the past a 4- ...
  3. S

    Reduction formula regarding binomial (1+x^2)^n

    Homework Statement prove the following reduction formula, n>0 ∫((1+x^2)^n) dx=(x(1+x^2)^n)(1/(2n+1)) +2n/(2n+1)∫(1+x^2)^(n-1) dx Homework Equations none The Attempt at a Solution one of many attempts, i get close, but no cigar. Huge blow to the calculus ego. Any help would be greatly...
  4. B

    Is My Attempt at Solving the Binomial Expansion Homework Correct?

    Homework Statement http://i47.tinypic.com/29o3ehc.png Homework Equations (1+x)^n=[1+nx/1!+(n)(n-1)x^2/2!+...+b^n] The Attempt at a Solution here's my attempt to part (i) http://i47.tinypic.com/2yv6y3s.png is it correct?
  5. X

    Probability distributions binomial or hypergeometric

    Homework Statement A committee of 16 persons is selected randomly from a group of 400 people, of whom are 240 are women and 160 are men. Approximate the probability that the committe contains at least 3 women. I just want to know if it's hyper geometric or binomial. I suspect it's hyper...
  6. Z

    How Many Edges and Keys in Specific Binomial Min-Heaps?

    Homework Statement a) What is the exact number of edges in a Binomial Min-Heap with 255 nodes? b) Consider a Binomial Min-Heap with 4097 distinct keys. What is the exact maximum number of keys that must be examined to find the minimum key? Homework Equations The Attempt at a...
  7. L

    Legendre polynomials and binomial series

    Homework Statement Where P_n(x) is the nth legendre polynomial, find f(n) such that \int_{0}^{1} P_n(x)dx = f(n) {1/2 \choose k} + g(n)Homework Equations Legendre generating function: (1 - 2xh - h^2)^{-1/2} = \sum_{n = 0}^{\infty} P_n(x)h^n The Attempt at a Solution I'm not sure if that...
  8. O

    Binomial expansion question that I cannot fathom

    Homework Statement It says: Determine the coefficient of p4q7 in the expansion of (2p-q)(p+q)10. I can find the coefficient of p4q6 in the expansion of (p+q)10 but how am I to find it for (2p-q)(p+q)10? Homework Equations Binomial expansion formula. The Attempt at a Solution...
  9. S

    How to calculate 200C65 (for binomial distribution formula)

    Hi, I have tried to calculate 200C65 on my calculator but the calculator gives an error. Do u know how to do it? I also tried to calculate it through the formula with the ! but doesn't give an answer.
  10. P

    Binomial expansion validity

    http://www.examsolutions.net/maths-revision/core-maths/sequences-series/binomial/formula/validity/tutorial-1.php On the above video, he states that the binomial expansion is only valid for |a| < 1 when n is not a positive integer. I understand that when n is not a positive integer the...
  11. Biosyn

    How Do You Calculate a Binomial Distribution Problem?

    Homework Statement Find the value of Ʃn(18 n)(0.46)^2(0.54)^(18-n) The sum is from n = 0 to n=18 Sorry, I do not know how to format it. Homework Equations I am using the Binomial Expansion Theorem: The Attempt at a Solution Not sure where to start. P = 0.46 Q =...
  12. B

    Why Does Calculating P(|Y-5| >= 3) Involve Y=7 in a Binomial Distribution?

    IF Y~B(11, 0.3), find (|Y-5| >= 3) I got the answer(0.3170) but i don't understand the logic behind this part where i am confused. can someone explain the working(second working) where i somehow got it blindly correct? ================================== my working at first: |Y-5|...
  13. D

    Calculating Binomial Distribution with a Calculator

    Homework Statement Hello, I am trying to calculate the following: 15!/(1!)(14!) x (0.80)^14 x (0.2)^1 I understand the problem as I have already put the numbers together. My trouble is actually using the calculator to find the answer. When I try to find 15! = 1.307674368^12 I am confused...
  14. H

    Binomial and Hypergeometric Distributions

    Homework Statement We have an urn with 5 red and 18 blues balls and we pick 4 balls with replacement. We denote the number of red balls in the sample by Y. What is the probability that Y >=3? (Use Binomial Distribution) Homework Equations The Attempt at a Solution Okay, so we...
  15. S

    Sum of binomial random variables

    Homework Statement let y_1 and y_2 be iid bin(5,1/4) random variables let v=y_1+2*y_2 and u = 3*y_1 -2y_2 find f_uv (u,v) and the cov(u,v) Homework Equations f_y (y) = (5 choose y) (1/4)^y (3/4)^5-x for x=0,1,2,3,4,5 covariance=E(uv)-E(u)E(v) The Attempt at a Solution...
  16. J

    Finding expected value of a Binomial

    Homework Statement I'm trying to solve the following question: You and n other people (so n+1 people) each toss a probability-p coin, with 0<= P \ <=1. Then each person who got a head will split some arbitrary amount of prize money, K, equally. If nobody gets a head, then nobody gets the prize...
  17. P

    Looking for an idea for proving inequality, probably using binomial theorem.

    Guess what? I just got my new calculus book last week! ^^ The book opens with the definition of the real numbers by Dedekind and goes to prove properties of this numbering system such as The supremum axiom and others. At the end of the chapter are about 30 exercises without their solutions...
  18. Z

    Information Theory- DMS question- Binomial dist?

    Homework Statement Let X1, . . . ,Xn be a message from a memoryless source, where Xi are in A. Show that, as n →∞, the proportion of messages in the typical set converges to zero, unless Xi is uniform on A. Homework Equations The Attempt at a Solution Confused, possibly because...
  19. J

    Trying to prove this equality involving a summation of a binomial coefficient.

    I immediately thought of induction, so that is what I used, but I can't seem to make any progress past a certain point.
  20. D

    Problem Related to Binomial Coefficient

    Hi All, Homework Statement This is algebraic proof of Vandermonde's identity: I am having some problem understanding how we reached the second last step and more importantly, last steps from revious steps. src::proofwiki.org I would be grateful if someone would elaborate it clearly...
  21. G

    How Does the Binomial Coefficient Calculate Combinations?

    Homework Statement Homework Equations The Attempt at a Solution do you see where it says 5 over 2 = 10 and 5 over 3 = 10. How? I don't get what they're doing.
  22. C

    Binomial Distribution with non integer succes

    I am doing a problem where I am to determine the probability that the number of students wanting a new book is within two standard deviations of the mean. μ +- 2δ comes out with a non integer number, in which I have to use to find probability. The equation to find probability uses the factorial...
  23. I

    MHB How to Solve a Problem Using Binomial Distribution and Normal Table?

    I have an assignment which is a bit different, I have to use Mathematics Handbook for Sience and Engineering to solve the problem, I can look it up in tables. But the tables for binomial functions is only up to 20, Normal Distribution to 3.4 and Poisson up to 24 in some cases. So how do I do...
  24. V

    Expressing the binomial coefficients

    Homework Statement Expressing the binomial coefficients in terms of factorials and simplifying algebraically, show that (n over r) = (n-r+1)/r (n over r-1);Homework Equations The Attempt at a Solution I honestly don't even know how to come about this problem...I really need help in this...
  25. srfriggen

    Binomial Theorem coefficients proof

    Homework Statement Define (n k) = n!/k!(n-k)! for k=0,1,...,n. Part (b) Show that (n k) + (n k-1) = (n+1 k) for k=1,2,...n. Part (c) Prove the binomial theorem using mathematical induction and part (b). Homework Equations The Attempt at a Solution I'm wasn't able to...
  26. A

    Prove binomial series converges for |x|<1

    Homework Statement How do you prove that the binomial series (x+1)^p converges for |x|<1 ? Homework Equations The Attempt at a Solution (x+1)^p = Ʃx^{n}\frac{p!}{(p-n)!n!} After doing ratio test I get |x|<1 . But now I have to test end points and this is my problem: when...
  27. D

    Proving n^n > 2^n *n using the Binomial theorem

    Homework Statement Prove that n^n > 2^n * n! when n > 6 using the Binomial theorem. I just proved the Binomial theorem using induction which was not that difficult but in applying what I learned through it's proof I am having difficulty. Homework Equations Binomial theorem = (x+y)^n =...
  28. T

    Binomial Distribution: Finding Probability with Trials, Success, and X Value

    Homework Statement I've uploaded a picture of the question. I need help in identifying the correct number of trials, probability of success and the X value(number of successes) Homework Equations i'm using the binomial distribution function on the calculator but I've attached the formula just...
  29. fluidistic

    Characteristic function of the binomial distribution

    Homework Statement Hey guys, I'm self studying some probability theory and I'm stuck with the basics. I must find the characteristic function (also the moments and the cumulants) of the binomial "variable" with parameters n and p. I checked out wikipedia's article...
  30. J

    Why does Binomial dist. converge in distribution to Poisson dist. ?

    Hey guys, In class, I was shown that the Binomial prob density function converges to the Poisson prob density function. But why does this show that the Binomial distribution converges in distribution to the Poisson dist. ? Convergence in distribution requires that the cumulative density...
  31. V

    MHB Sum with Binomial Coefficients

    I'm having trouble proving the following identity (I don't even know if it's true): $$\sum_{r=1}^k \binom{k}{r} \binom{n-k-1}{r-1}=\binom{n-1}{k-1}$$ $$\forall n,k \in \mathbb{N} : n>k$$ Thank you in advance for any help! Vincent
  32. S

    Cumulative distribution of binomial random variables

    Homework Statement The probability of being dealt a full house is approximately 0.0014. Find the probability that in 1000 hands of poker you will be dealt at least 2 full houses Homework Equations I can use binomial distribution. The Attempt at a Solution The probability of getting...
  33. O

    Variance of binomial distribution - 1 trial

    Homework Statement For n trials, S_n can be seen as the sum of n independent single trials X_i, i = 1,2,...,n, with \mathbb{E}[X_i]=p and Var[X_i]=p(1-p).2. What I don't understand I don't understand why Var[X_i]=p(1-p). We know that: Var[X_i]=\mathbb{E}[(X_i - \mathbb{E}[X_i])^2] =...
  34. sankalpmittal

    Question regarding binomial theorem.

    Homework Statement (√2 + 1)6 = I + f Where I is the sum of integer part of the expansion of (√2 + 1)6 and f is sum of the fraction part in (√2 + 1)6. Homework Equations (x+1)n = nC0 xn + nC1 xn-1 + nC2 xn-2 + ... + nCn nCn = nC0 = 1 The Attempt at a Solution I expanded...
  35. J

    Mastering Binomial Theorem for Understanding Rudin's Analysis Proofs

    I have been teaching myself analysis with baby rudin. I have just started chapter three in the past week or so and one thing I am having trouble with is the proofs which use the binomial theorem and various identities derived from it. Rudin pretty much assumes this material as prerequisite and...
  36. U

    Binomial Theorem proof by induction, Spivak

    Homework Statement Prove the binomial theorem by induction. The attempt at a solution http://desmond.imageshack.us/Himg35/scaled.php?server=35&filename=sumu.png&res=landing Hi, running into trouble with this proof and google hasn't helped me. I don't understand the jump here, and as...
  37. C

    Probability Finding Binomial Average Problem

    Homework Statement In each batch of manufactured articles, the proportion of defective articles is p. From each batch, a random sample of nine is taken and each of the nine articles is examined. If one article is found to be defective, the batch is rejected; otherwise, it is accepted. If...
  38. G

    Binomial Theorem Homework Help: 3/2 in Parentheses?

    Homework Statement What am I supposed to do with the 3 over 2 in the parentheses? It can be divide and it can be take the factorial. So what do I do with it?
  39. N

    Binomial Theorem: Find Expansion & Approximation of 97^(1/2)

    Homework Statement Find the first four terms in the expansion of \left(1-3x\right)^{3/2}. By substituting in a suitable value of x, find an approximation to 97^{1/2}. Homework Equations The Attempt at a Solution I used the binomial expansion formula to work the answer and it is 1-...
  40. O

    Small question about binomial theorem

    I was trying to make sense of the equation attached below which was on the wikipedia site. However I'm not entirely sure how to make use of the "n choose 0" , "n choose 1", etc. statements that in front of each term in of the expansion. I roughly know how the expansion should look...
  41. T

    Solve Binomial Theorem: Find Term Independent of x

    Homework Statement The method of Binomial expansion is useful because you can avoid expanding large expressions: Q: Find the term indepedent of x in the expansion of (2+x)[2x+(1/x)]5 The attempt at a solution: "For this to produce a term independent of x, the expansion of [2x+(1/x)]5 must...
  42. L

    How Can You Use Pascal's Triangle to Find the Binomial Series of (3+x)3?

    how to do binomial series of (3+x)3.
  43. W

    Coin Flipping: Binomial Distribution and Expected Product

    Question is: "If you roll a fair coin 10 times what is the expected product of number of heads and number of tails?" Someone answered 25 at at glassdoor.com. My answer would be: E(k(10-k)) where k is the rv representing the number of heads thrown. = 10E(k) - E(k^2) = 10*mean - (var +...
  44. A

    Binomial Distribution: Find p, given CDF

    I have a question about binomial distribution There is a random var X follows Binomial distribution ~B(n,p), where n is known but p is UNKNOWN. It is also known that a for known value of x, CDF(x) = Pr(X<=x) = 0.9 Is there anyway to estimate p? To give a concrete example, if n=8...
  45. agnibho

    Binomial theorem problem on the terms of an expansion

    Homework Statement Find an approximation of (0.99)5 using the first three terms of its expansion. 2. The attempt at a solution To get to the binomial theorem I divided 0.99 into (0.99)5 = (1-0.01)5 = {1+(-0.01)}5 Then, T1 = 5C0(1)5 = 1 x 1=1 T2 = 5C1(1)5-1(-0.01)1 = 5x1x...
  46. perplexabot

    Binomial Expansion: Solve for L=Hbar*(l+1/2)

    Hey all. I have posted a thread regarding this question a while back. I did get an answer and everything. (Here is my old post along with the original question if you are interested: https://www.physicsforums.com/showthread.php?t=592885). So i tried doing that problem again like this: Given...
  47. Y

    Finding a Minumum N from Binomial Distribution

    Homework Statement From the text: Use Hershey's Kisses to estimate the probability that when dropped, they land with the flat part lying on the floor. How many trials are necessary to get a result that appears to be reasonably accurate when rounded to the first decimal place? Homework...
  48. J

    MHB Binomial Integral for Non-Negative Integer $n$

    for a non nagative integer $n$, If $\displaystyle I_{n}=\int_{0}^{1}\binom{x}{n}dx$, then $I_{n}=$ where $\displaystyle \binom{n}{r} = \frac{n!}{r!.(n-r)!}$
  49. C

    Use Binomial Theorem and appropriate inequalities to prove

    Use Binomial Theorem and appropriate inequalities to prove! Homework Statement Use Binomial Theorem and appropriate inequalities to prove 0<(1+1/n)^n<3 Homework Equations The Attempt at a Solution So I started by.. \sum ^{n}_{k=0} (n!/(n-k)! k!) a^{n-k}b^{k} = n!/(n-k)!k! (1)^{n-k}...
  50. U

    Chi-Squared Test on a table of Binomial Variates - Finding the expected frequencies

    Hi, Carry out a chi-squared test for the following table of frequencies of X ∼ Binomial(5,p) variates when (a) p = 0.3 x 0 1 2 3 4 5 Observed 162 346 303 149 36 4 frequency Now I know how to carry out the chi-squared test once I have...
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