A+ » VCE » Maths Methods U3 & 4 Master Notes » A4 Probability and Statistics » Binomial Distribution

# 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.
Read More »3.7 Normal Approximation to the Binomial Distribution

## 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.
Read More »2.5 Properties of the Binomial Distribution

## 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

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:
Read More »2.4 Bernoulli Sequence and Binomial Distribution

## The Binomial Distribution [Video Tutorial]

This tutorial covers material encountered in chapter 14 of the VCE Mathematical Methods Textbook, namely: Bernoulli distribution Binomial distribution Mean and variance of a binomial… Read More »The Binomial Distribution [Video Tutorial]