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.

View More On Wikipedia.org
Recent contents Top users
  1. S

    For what value of θ is the binomial probability b(x;n,θ) maximized?

    If X is a binom. rand. var., for what value of θ is the probability b(x;n,θ) at max? Ive no idea... My only guess (most likely wrong) is that max and min are always derivatives... So do i just differentiate and express θ...? Any suggestions...?=( Thank you!
  2. ArcanaNoir

    Find the MGF of geometric,neg binomial dist.

    Homework Statement Find the MGF (Moment generating function) of the a. geometric distribution b. negative binomial distribution Homework Equations geometric distribution: f(x)=p^x(1-p)^{x-1} where x=1,2,3... negative binomial distribution: f(x)= \frac{(x-1)!}{(x-r)!(r-1)!}p^r(1-p)^{x-r}...
  3. S

    Description of binomial expansion

    what will be the coefficient of the x^n in the expansion of (1+x)^-1 and(1-x)^-1. Please answer it separately..
  4. M

    Probability of <3 Errors in 10-char Msg

    In sending 10 characters, a character error occurs independently with probability 1/10. What is the probability that in a 10-character message, less than 3 errors occur? I am using the binomial formula (n choose k)pk(1-p)n-k where n = 10, p = 1/10, and k is the number of errors. Since the...
  5. K

    Proof using the Binomial Theorem

    Homework Statement Homework Equations The Attempt at a Solution I am really stuck, I have no clue how to even begin. For part B I tried changing the RHS to factorials but I was left at a dead end there.
  6. J

    Proving Binomial Sums: Step-by-Step Guide for Solving Homework Equations"

    Homework Statement https://www.physicsforums.com/attachment.php?attachmentid=39642&stc=1&d=1317853920 how do you go about solving this? Homework Equations i have proved the binomial theorem.The Attempt at a Solution i was considering cases, for j(even or odd). would this be the right direction?
  7. S

    Negative Binomial and Chi-square

    can anyone help me with this question :
  8. S

    Probabilities and binomial theorem

    Homework Statement Consider an ideal gas of N identical particles in a volume V, and a subvolume v. The chance a molecule is in inside the subvolume is P = v/V. a) What is the chance the subvolume contains n particles? b) Use the binomial theorem (p + q)^N = \sum_{n = 0}^N p^n q^{N-n}...
  9. S

    Negative Binomial random variable

    Data is collected on the number of fish caught per day on a month long fishing expedition. It is hypothesised that the data are consistent with a negative Binomial random variable ,X , starting at 0, so that X~Neg Bin(k,p) where E[X]=k(1-p)/p and Var =k(1-p)/p^2 . However, before a hypothesis...
  10. R

    Can Stirling's Formula Derive the Normal Distribution from the Binomial?

    I want to show that the binomial distribution: P(m)=\frac{n!}{(n-m)!m!}p^m(1-p)^{n-m} using Stirling's formula: n!=n^n e^{-n} \sqrt{2\pi n} reduces to the normal distribution: P(m)=\frac{1}{\sqrt{2 \pi n}} \frac{1}{\sqrt{p(1-p)}} exp[-\frac{1}{2}\frac{(m-np)^2}{np(1-p)}]...
  11. K

    Probability of at Least 14 Not Having ETFs in 25 Investor Portfolios

    Hi, the following is a list of binomial cumulative distribution of the probability that out of 25 investors, the number of investors that would have exchange-traded funds in their portfolios. We were asked for the probability that at least 14 investors do not have exchange-traded funds in...
  12. P

    How do I simplify a binomial division?

    Homework Statement (v-4)/(5v+1) The Attempt at a Solution I'm an engineering student, and I'm taking Differential Equations, but I can't remember how to do simple things like this. A walk through explanation would be very much appreciated, I don't have a lot of time to spare. The...
  13. P

    Proof of Binomial Identity: Proving SUM(nCk)*2^k=(3^n+(-1)^n)/2

    Homework Statement Prove that for all positive integers n, the equality holds: SUM(nCk)*2^k=(3^n+(-1)^n)/2 Note: The sum goes from k=0 to n. AND k has to be even. Homework Equations Binomial Theorem The Attempt at a Solution I know that if we use the binomial theorem for x=2 and...
  14. E

    Binomial Probability Problems: Finding Probability for Glasses and DMF Teeth

    Problem 1: About 50% of all persons age 3 and older wear glasses or contact lenses. For a randomly selected group of five people find the probability that: a. exactly three wear glasses or contact lenses b. at least one wears them c. at most one wears them For this problem I set n=5...
  15. W

    Joint hipergeometric and binomial probability?

    Have an example: 123455 111555 the sample space is 0..9 row 1 can pick 6 numbers out of which 2 are repeated row 2 can pick 6 numbers out of which 3 by 3 are repeated I want to know what is the real probability that row 1 will match with 2 distinct numbers numbers row 2, and a repeated number...
  16. P

    Binomial Expansion alternating Sum

    use the binomial expansion formula to find the alternating sum of the numbers in row n of Pascals triangle. nC0-nC1+nC2... So I wrote a few rows of the triangle and it looks like once you get past row zero the alternating sum of the numbers in the rows all add to zero. Does that work...
  17. L

    Variance (error bars) with a binomial proportion

    I have a list of chemicals, their assay test results, and a binomial column of whether or not the assay test result was high enough to be considered a threat (anything >2g/ml). Some chemicals were tested more than once, but others were not. It is understood that it is a poor set of data, but I...
  18. J

    Newton's Binomial Theorem to Estimate, Find Error

    Homework Statement Use Newton's Binomial Theorem to estimate integral of (1+x^4)^(1/2) from 0 to 1/2 to within one part in 1000, (error>1/1000) Homework Equations I used the Binomial Series expansion, so (a+b)^n = a^n +na^(n-1)b + (n(n-1))/2! (etc The Attempt at a Solution I...
  19. E

    Understanding Binomial PMF Notation: Clearing Up Confusion?

    Hi, could someone possible make something clear for me - I have come across this notation for a binomial PMF formed from an underlying beurnolli distribution: PS_{n}(\bar{p}n)\sim\sqrt{\frac{1}{2\pi n\bar{p}(1-\bar{p})}}exp [n\varnothing(p,\bar{p}] ,\\...
  20. Rasalhague

    Poisson & normal distributions as approximations for the binomial

    These three quotes talk about the use of the Poisson and normal distributions as approximations for the binomial when n is large. The first two quotes here say Poisson is best when p small, and the normal otherwise. The third seems to change the story; it says Poisson is best for large p too. Is...
  21. Rasalhague

    What is the relation between probability spaces and the binomial distribution?

    Here, to further test my understanding, is an attempt to apply the measury theory definitions of a probability space to the binomial distribution. All comments welcome! Let (R,D,O) be a probability space: R = \left \{ 0,1 \right \} D = 2^R O:D\rightarrow[0,1] \; | \; O(\left \{ 1...
  22. Q

    Where Did the Binomial Theorem Originate?

    Where did the binomial theorem come from...?
  23. F

    Sum of binomial coefficients and cos(kx)

    Homework Statement Calculate the following sum: (click to expand)The Attempt at a Solution I tried something with Moivre formula and Newton binomial theorem but no result :redface:, should i continue with these or is there any simpler approach? I just need some hints. Thanks.
  24. B

    From multinomial distribution to binomial distribution

    Homework Statement (N1, ... , Nr) has multinomial distribution with parameters n and p1, ... , pr. Let 1 \leq i < j \leq r. I am looking for an intuitive explanation for the 3 following questions. a) What is the distribution of Ni? b) What is the distribution of Ni + Nj? c) What is the...
  25. noowutah

    Divisible binomial coefficients

    Homework Statement I need to sum the binomial coefficients that are divisible by a positive integer t, i.e. \sum_{i=0}^{s}\binom{ts}{ti} Is there any way to get rid of the sum sign? Homework Equations Let t be fixed and s go to (positive) infinity (both t and s are positive...
  26. T

    A binomial problem involving 2 different random variables.

    In a recent federal appeals court case, a special 11-judge panel sat to decide on a certain particular legal issue under certain particular facts. Of the 11 judges, 3 were appointed by political party A, and 8 were appointed by political party B. Of the party-A judges, 2 of 3 sided with the...
  27. P

    How Do You Solve a Binomial Expansion Problem with Given Series Terms?

    Hi, When (1+ax)n is expanded as a series in expanding powers of x, the first three terms are 1 - 8x + 30x2 Calculate the values of a and n. So, I think we need simultaneous equations and I managed to build the first one: a*n = 8 My problem is however, that to construct the second...
  28. Saitama

    Binomial Theorem: Is 11n+2 + 122n+1 divisible by 113, 123, or 133 for n in N?

    Homework Statement If n \in N, then 11n+2 + 122n+1 is divisible by:- a)113 b)123 c)133 Homework Equations The Attempt at a Solution I did it by substituting different values of n and divided by each of the option. Answer came out to be 133. But I want to do it step by step...
  29. J

    Binomial coefficients sum conjecture about exponential

    Fix some constant 0<\alpha \leq 1, and denote the floor function by x\mapsto [x]. The conjecture is that there exists a constant \beta > 1 such that \beta^{-n} \sum_{k=0}^{[\alpha\cdot n]} \binom{n}{k} \underset{n\to\infty}{\nrightarrow} 0 Consider this conjecture as a challenge. I don't...
  30. S

    Probability of Binomial Variable ≥ Another Binomial Variable

    If two binomially distributed variables are generated as paired events, how often will the variable with p=X be greater than the variable with p=Y? Also what is the "equity" if ties are counted as .5 for each? For instance in Excel I generated 10,000 numbers with p=.8 and 10,000 with p=.6...
  31. B

    Transforming a uniform distribution into a binomial

    Homework Statement Let X~UNIF(0,1). Find y = G(u) such that Y = G(U)~BIN(3,1/2) Homework Equations The Attempt at a Solution after a bit of searching/reading, i found how to do this with a continuous distribution (the problem i had was an exponential, so i took the inverse)...
  32. I

    Power and binomial distribution

    Maybe someone is really good with stats, or has access to a statistics professor. Here we go: I am trying to determine the power for a study. The distribution is binomial. I have a device that either works or does not work. I do not know the real probability, but I think it is very good...
  33. A

    Uncertainty for p = 0 for binomial distribution?

    I have some data (4 runs each of about 10 trials) which is binomial with n_hits/N_trials n/N = 0/11, 0/9, 0/10, 0/10 So, I estimate the probability p = n/N = 0 But how can I calculate an uncertainty on this value? I thought to try total N_tot=40 and n_tot=1, so p_tot=1/40 = 0.025 (i.e...
  34. X

    Coefficient of x^35 in Binomial Theorem Expansion

    Homework Statement Find the coefficient of http://webwork2.math.utah.edu/webwork2_files/tmp/equations/73/3e29a3b979c709dbb6c609c5a6ce891.png in the expansion of [PLAIN][PLAIN]http://webwork2.math.utah.edu/webwork2_files/tmp/equations/63/dcb58790e8122dce61b830977294091.png Homework Equations...
  35. O

    Conditional Probability and/or Binomial Distribution(s)

    Homework Statement A car dealer estimates that 50% of customers entering the dealership will buy a normal car, 20% will buy a high-end car, and 30% are just browsing. If 5 customers enter his dealership on a particular day, what is the probability that two will purchase high-end models, one...
  36. coolul007

    Modulus Predictions for Binomial Expansion Coefficients?

    I am trying to predict the modulus without really doing the expansion. Therefore I'm in a snag with actually computing vs. only computing what I think I need. Here's the assumption I am Making: n C r == 0 mod (n-1) for all r > 1 n C r are the coefficients of the binomial expansion. My...
  37. K

    Expected value of function in binomial distribution

    Hi members, Hope someone can help with this assignment question? I need to proof: E(1/1+X) = [1-(1-p)^n+1]/p(n+1) where X ~ Bi(n,p) Below are my steps and I'm not sure where I went wrong: 1. sum(x=0 to n) (1/1+x)*(n choose x)*p^x*(1-p)^n-x 2. sum(x=0 to n)...
  38. B

    Electric Dipole and Electric Potential and binomial approximation

    Electric Dipole and Electric Potential.. and binomial approximation! Homework Statement An electric dipole at the origin consists of two charges +q and -q spaced distance s apart along the y-axis. a.)Find an expression for the potential V(x,y) at an arbitrary point in the xy-plane...
  39. K

    Expanding Binomials: Simplifying Complex Expressions

    Homework Statement by expanding the binomial show that ( (Sqtroot(3)/2) + (1/2)i )^4 = ( (-1/2) + (sqroot(3)/2)i ) The Attempt at a Solution I'm stuck, I now ( (Sqtroot(3)/2) + (1/2)i )^4 = ( (9/16) + (1/16)i ) But that's all I got, don't know the next steps.
  40. G

    Partial Binomial Expansions, and acceptable notation.

    What is acceptable summation notation for a binomial expansion of, for example (1+x)^n, from the zeroth to the (n-1)th term? For example a possible expansion maybe (1+x)^4, where by I would like to write in summation notation that the expansion would be : 1 + 4x + 6(x^2) 4(x^3) . Notice...
  41. M

    Approximating the Square Root of 11 Using Binomial Expansion

    Homework Statement Prove that , if x is so small that terms in x3 and higher powers may be neglected, then http://www.mathhelpforum.com/math-help/attachments/f37/21081d1299568824-binomial-expansion-question-msp520319eed9935g5h93hf000055b8ic10787h09a9.gif . By substituting a suitable value of x...
  42. M

    Binomial Expansion: Understanding the Coefficient of x5

    Homework Statement Find the coefficient of x5 for the expression: (1-x)6(2x+3)4 The answer provided is -222, but my answer is far from that, can any enlighten me? Homework Equations The Attempt at a Solution
  43. K

    About random variable and Binomial distribution

    Hi there, As many texts' discussion, we usually use a variable x for any value randomly picked. For a Bernoulli trials, i.e. each random variable x can either be successful or fail. If the probability of success if p and that of failure is q=1-p, then the expectation value of x would be...
  44. G

    Help Binomial Distribution: Statistics for M.E's

    Help! Binomial Distribution: Statistics for M.E's Homework Statement Four wheel bearings are to be replaced on a company vehicle. The mechanic has selected the four replacement parts from a large supply bin in which 10% of the bearings are defective and will fail within the first 100 miles...
  45. mnb96

    Characteristic function of binomial distribution.

    Hello, I considered a Binomial distribution B(n,p), and a discrete random variable X=\frac{1}{n}B(n,p). I tried to compute the characteristic function of X and got the following: \phi_X(\theta)=E[e^{i\frac{\theta}{n}X}]=(1-p+pe^{i\theta/n})^n I tried to compute the limit for n\to +\infty...
  46. S

    Characteristic equation of binomial random variable

    Homework Statement find the characteristic equation of a binomial variable with pmf p(x) =\frac{n!}{(n-k)!k!}*p^{k}*(1-p)^{n-k}Homework Equations characteristic equation I(t) = \sump(x)*e^{tk}The Attempt at a Solution I(t) = \sum\frac{n!}{(n-k)!k!}*(p^{k}*(1-p)^{-k}*e^{tk})*(1-p)^{n} i am...
  47. Z

    Exploring the Binomial Series: A Comprehensive Homework Statement

    Homework Statement The Attempt at a Solution Is there any difference between the above expression and ? Is there any relation between these two?
  48. A

    Engineering Statistics: Binomial Distribution

    Homework Statement Let k >= 3 be any integer. What is the probability that a random k-digit number will have at least one 0, one 1 and one 2? (as usual every number starts with either 1,2,...9 and NOT zero) Homework Equations b(x : n,p) = (n x)p^x*(1-p)^(n-x) where x = 0, 1, 2, ... ,n...
  49. R

    Binomial expansion of massive spin 0 propagator

    I've seen written the following manipulation of the KG Feynman propagator: \frac{1}{p^2-m^2+i\epsilon}=\frac{1}{p^2+i\epsilon} \frac{1}{(1-\frac{m^2}{p^2+i\epsilon})}= \frac{1}{p^2+i\epsilon} (1+\frac{m^2}{p^2+i\epsilon}+\left(\frac{m^2}{p^2+i\epsilon}\right)^2+...) I don't think this...
  50. Q

    What is the Probability of Fibrillation After N Attempts?

    Homework Statement The question provides a table and asks: Number of Attempts Fraction persisting in fibrillation 0 1.00 1 0.37 2...
Back
Top