binomial

(noun)

A polynomial with two terms.

Related Terms

  • binomial coefficient
  • integer
  • square
  • Monomials
  • Binomials
  • Multiplication of Two Monomials
  • Binomial
  • monomial
  • polynomial
  • trinomial

(noun)

A polynomial consisting of two terms, or monomials, separated by addition or subtraction symbols.

Related Terms

  • binomial coefficient
  • integer
  • square
  • Monomials
  • Binomials
  • Multiplication of Two Monomials
  • Binomial
  • monomial
  • polynomial
  • trinomial

(noun)

A polynomial consisting of two terms, or monomials, separated by an addition or subtraction symbol.

Related Terms

  • binomial coefficient
  • integer
  • square
  • Monomials
  • Binomials
  • Multiplication of Two Monomials
  • Binomial
  • monomial
  • polynomial
  • trinomial

Examples of binomial in the following topics:

  • Mean, Variance, and Standard Deviation of the Binomial Distribution

    • In this section, we'll examine the mean, variance, and standard deviation of the binomial distribution.
    • The easiest way to understand the mean, variance, and standard deviation of the binomial distribution is to use a real life example.
    • In general, the mean of a binomial distribution with parameters $N$ (the number of trials) and $p$ (the probability of success for each trial) is:
    • $s^2 = Np(1-p)$, where $s^2$ is the variance of the binomial distribution.
    • Coin flip experiments are a great way to understand the properties of binomial distributions.
  • Additional Properties of the Binomial Distribution

    • In this section, we'll look at the median, mode, and covariance of the binomial distribution.
    • There are also conditional binomials.
    • The binomial distribution is a special case of the Poisson binomial distribution, which is a sum of n independent non-identical Bernoulli trials Bern(pi).
    • This formula is for calculating the mode of a binomial distribution.
    • This summarizes how to find the mode of a binomial distribution.
  • Binomial Expansions and Pascal's Triangle

    • The binomial theorem, which uses Pascal's triangles to determine coefficients, describes the algebraic expansion of powers of a binomial.
    • The binomial theorem is an algebraic method of expanding a binomial expression.
    • This formula is referred to as the Binomial Formula.
    • Applying these numbers to the binomial expansion, we have:
    • Use the Binomial Formula and Pascal's Triangle to expand a binomial raised to a power and find the coefficients of a binomial expansion
  • The normal approximation breaks down on small intervals

    • Caution: The normal approximation may fail on small intervals The normal approximation to the binomial distribution tends to perform poorly when estimating the probability of a small range of counts, even when the conditions are met.
    • However, we would find that the binomial solution and the normal approximation notably differ:
    • We can identify the cause of this discrepancy using Figure 3.19, which shows the areas representing the binomial probability (outlined) and normal approximation (shaded).
    • TIP: Improving the accuracy of the normal approximation to the binomial distribution
    • The outlined area represents the exact binomial probability.
  • Complex Numbers and the Binomial Theorem

    • Powers of complex numbers can be computed with the the help of the binomial theorem.
    • Recall the binomial theorem, which tells how to compute powers of a binomial like $x+y$.
    • Using the binomial theorem directly, this can be written as:
    • Recall that the binomial coefficients (from the 5th row of Pascal's triangle) are $1, 5, 10, 10, 5, \text{and}\, 1.$ Using the binomial theorem directly, we have:
    • Connect complex numbers raised to a power to the binomial theorem
  • Multiplying Algebraic Expressions

    • Multiplying two binomials is less straightforward; however, there is a method that makes the process fairly convenient.
    • "FOIL" is a mnemonic for the standard method of multiplying two binomials (hence the method is often referred to as the FOIL method).
    • Outer (the "outside" terms are multiplied—i.e., the first term of the first binomial with the second term of the second)
    • Inner (the "inside" terms are multiplied—i.e., the second term of the first binomial with the first term of the second)
    • Remember that any negative sign on a term in a binomial should also be included in the multiplication of that term.
  • The Binomial Formula

    • The binomial distribution is a discrete probability distribution of the successes in a sequence of $n$ independent yes/no experiments.
    • This makes Table 1 an example of a binomial distribution.
    • The binomial distribution is the basis for the popular binomial test of statistical significance.
    • However, for $N$ much larger than $n$, the binomial distribution is a good approximation, and widely used.
    • Is the binomial coefficient (hence the name of the distribution) "n choose k," also denoted $C(n, k)$ or $_nC_k$.
  • The Normal Approximation to the Binomial Distribution

    • The process of using the normal curve to estimate the shape of the binomial distribution is known as normal approximation.
    • The binomial distribution can be used to solve problems such as, "If a fair coin is flipped 100 times, what is the probability of getting 60 or more heads?"
    • The process of using this curve to estimate the shape of the binomial distribution is known as normal approximation.
    • The normal approximation to the binomial distribution for 12 coin flips.
    • Note how well it approximates the binomial probabilities represented by the heights of the blue lines.
  • Normal approximation to the binomial distribution

    • We might wonder, is it reasonable to use the normal model in place of the binomial distribution?
    • Here we consider the binomial model when the probability of a success is p = 0.10.
    • Showing that the binomial model is reasonable was a suggested exercise in Example 3.50.
    • With these conditions checked, we may use the normal approximation in place of the binomial distribution using the mean and standard deviation from the binomial model: µ = np = 80 σ =$\sqrt{np(1p)}$ = 8.
    • Hollow histograms of samples from the binomial model when p = 0.10.
  • Binomial Expansion and Factorial Notation

    • The binomial theorem describes the algebraic expansion of powers of a binomial.
    • Recall that the binomial theorem is an algebraic method of expanding a binomial that is raised to a certain power, such as $(4x+y)^7$.
    • The coefficients that appear in the binomial expansion are called binomial coefficients.
    • Example: Use the binomial formula to find the expansion of $(x+y)^4$
    • Use factorial notation to find the coefficients of a binomial expansion
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