binomial

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