binomial distribution

Examples of binomial distribution 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.
  • As with most probability distributions, examining the different properties of binomial distributions is important to truly understanding the implications of them.
  • The easiest way to understand the mean, variance, and standard deviation of the binomial distribution is to use a real life example.
  • $s^2 = Np(1-p)$, 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.
  • 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).
  • Usually the mode of a binomial B(n, p) distribution is equal to where is the floor function.
  • This formula is for calculating the mode of a binomial distribution.
  • This summarizes how to find the mode of a binomial distribution.

• 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.

• 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.
  • 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 is a good approximation, and widely used.

• History of the Normal Distribution

  • In the chapter on probability, we saw that the binomial distribution could 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?
  • Binomial distributions for 2, 4, 12, and 24 flips are shown in Figure 1.
  • Examples of binomial distributions.
  • 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.

• 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.
  • Notice that the width of the area under the normal distribution is 0.5 units too slim on both sides of the interval.
  • TIP: Improving the accuracy of the normal approximation to the binomial distribution
  • The normal approximation to the binomial distribution for intervals of values is usually improved if cutoff values are modified slightly.The cutoff values for the lower end of a shaded region should be reduced by 0.5, and the cutoff value for the upper end should be increased by 0.5.
  • The outlined area represents the exact binomial probability.

• Normal Approximation to the Binomial

  • State the relationship between the normal distribution and the binomial distribution
  • In the section on the history of the normal distribution, we saw that the normal distribution can be used to approximate the binomial distribution.
  • The binomial distribution has a mean of μ = Nπ = (10)(0.5) = 5 and a variance of σ2 = Nπ(1-π) = (10)(0.5)(0.5) = 2.5.
  • The problem is that the binomial distribution is a discrete probability distribution, whereas the normal distribution is a continuous distribution.
  • The difference between the areas is 0.044, which is the approximation of the binomial probability.

• Normal approximation to the binomial distribution

  • In some cases we may use the normal distribution as an easier and faster way to estimate binomial probabilities.
  • We might wonder, is it reasonable to use the normal model in place of the binomial distribution?
  • Figure 3.18 shows four hollow histograms for simulated samples from the binomial distribution using four different sample sizes: n = 10,30,100,300.
  • The approximate normal distribution has parameters corresponding to the mean and standard deviation of the binomial distribution: µ = np and σ = $\sqrt{np(1p)}$
  • 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.

• Binomial Probability Distributions

  • This chapter explores Bernoulli experiments and the probability distributions of binomial random variables.
  • The distribution of the number of successes is a binomial distribution.
  • Such a success/failure experiment is also called a Bernoulli experiment, or Bernoulli trial; when $n=1$, the Bernoulli distribution is a binomial distribution.
  • These probabilities are called binomial probabilities, and the random variable $X$ is said to have a binomial distribution.
  • A graph of binomial probability distributions that vary according to their corresponding values for $n$ and $p$.

• Student Learning Outcomes