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Binomial Distribution

3.7 Normal Approximation to the Binomial Distribution

Normal Approximation to the Binomial Distribution

  • Binomial distributions depend on n and p. We will observe that as n increases, if we plot the probability distribution of the binomial distribution, a ‘bell curve’ which is a symbolic shape of normal distribution, emerges. Below is the plot of the binomal distribution for n=10,\ 20,\ 50,\ 100 and p=0.2,\ 0.5,\ 0.8.
  • As you can see, as n increases, a smoother bell curve is seen. As for p, it only decides if the curve is slanted (or specifically ‘skewed’) to the left or right.
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2.5 Properties of the Binomial Distribution

Graph of the Binomial Distribution

  • Recall that a binomial distribution has the format

\operatorname{Pr}(X=x)=\left(\begin{array}{l}n \\x\end{array}\right) p^{x}(1-p)^{n-x} \quad \ \ \ \ \ \ \ \ \ x=0,1,2, \ldots, n

where X \sim \operatorname{Bi}(n, p).

  • We can plot the graph of (number of successes) against \operatorname{Pr}(X=x) \text { or } p(x) (corresponding probabilities) to see the graphical representation of the binomial distribution.
  • The graph of the binomial distribution will generally look like a small mountain. In other words, you would expect probability of number of successes to be higher when x is roughly half the number of trials.
  • However, it is obvious that if p is higher, then we would expect more successes. Therefore, the value of p will cause the graph to skew sideways. Refer to the example below to see what ‘positively/negatively skewed’ means graphically. To give a heads up, it means the graph would be slanted towards the left/right.
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2.4 Bernoulli Sequence and Binomial Distribution

The Bernoulli Sequence

  • There are many types of random variables that contributes to a discrete probability distribution. For instance, the number of ‘sixes’ obtained from 10 throws of a fair dice, and the amount of lottery winning are examples of a discrete random variables.
  • For a trial that has only either success or fail with a fixed probability, we call this as a Bernoulli trial. That is, the probability distribution table is given by
x\text { Outcome } 1 \text { (Success) }\text { Outcome } 2 \text { (Fail) }

where p is a constant.

  • Following this logic, we can also say that Bernoulli random variables are used to model two-outcome situations with fixed probability for each outcome.
  • In Bernoulli trial, we define success as x=1, and fail as x=0. Therefore, we can calculate the mean of a Bernoulli trial random variable as p, and its variance as p(1-p).
  • A particular expansion of this is the Bernoulli sequence. It describes a sequence of repeated trials with the following properties:
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