Theoretical Distributions

Introduction to Theoretical Distributions and Binomial Distribution

Objective

After going through this lesson, you shall be able to understand the following concepts.

• Meaning of Probability Distribution
• Binomial Distribution
• Mean, Mode and Variance of Binomial Distribution
• Fitting of Binomial Distribution

Introduction

There are two types of distributions in statistics: observed frequency distribution and probability distribution. In the previous chapter, we have learnt about observed frequency distribution. In this lesson and in the subsequent lessons, we will learn about probability distributions.

Meaning of Probability Distribution

Let us suppose we are performing an experiment by tossing a coin. What will we get? We will get either a head or a tail.
Now, if we know the probability of each outcome (i.e. head and tail), then we will get the probability distribution. The probability of both the outcomes is 0.5.
Hence, a probability distribution is a set of all possible outcomes of a probability experiment together with the probability of each outcome. A probability distribution can be discrete or continuous.
In a discrete probability distribution, the total probability gets distributed into different mass points, whereas in a continuous probability distribution, the total probability gets distributed into different class intervals.
Thus, formally, a probability distribution can be defined as a distribution that is drawn mathematically under certain probability laws. Since this type of distribution is based on expectations and exists in theory and in real life, it is also known as theoretical distribution or expected frequency distribution.

Theoretical probability distributions play a very important role in the statistical theory. The following note explains the importance of a theoretical probability distribution.



A theoretical probability distribution can be either a discrete probability distribution or a continuous probability distribution. This classification is based on the nature of the random variable.



Binomial Distribution

Binomial distribution is the most important and widely used discrete probability distribution. It is derived from a random experiment known as the Bernoulli process, which is named after the Swiss mathematician James Bernoulli. In this distribution, a random experiment is performed repeatedly under the same condition.
Suppose we roll a dice 50 times. Now, we need to find the probability of getting the number 5 exactly ten times.
How can we solve this problem?
Now, here comes the role of binomial distribution.
Before defining the binomial distribution, let us first discuss the Bernoulli trials.
Here, the dice is rolled 50 times, so the number of trials is 50, which is finite.
Now, the outcome of one trial will not affect the outcome of other trials.The trials are independent of each other.
In any of the trials, a head and a tail do not occur simultaneously, so the outcomes are mutually exhaustive.
Each trial results in two outcomes: either the number 5 occurs or any number other than 5 occurs. The probability of the occurrence of the number 5 remains the same in each trial.
So, the trials of a random experiment are called the Bernoulli trials if they satisfy the following conditions:
1.There is a finite number of trials.
2. The trials are independent and mutually exhaustive.
3. Each trial has two outcomes: success and failure.
4. The probability of success remains the same in each trial.

The probability distribution of the numbers of successes in an experiment consisting of n Bernoulli trials may be obtained by the binomial expansion of (q + p)ⁿ and this distribution of the numbers of successes X can be written as

X	0	1	2	…	X	…	n
P(X)				…		…

This probability distribution is known as the binomial distribution with parameters n and p, where n is the number of Bernoulli trials, p is the probability of success in one trial and q is the probability of failure in one trial which is 1 − p.

The probability of x successes P(X = x) is denoted by p(x) and is given by
$P\left(X=x\right)={}^{n}C_x p^x q^{n-x}, x=0, 1, 2, ..., n$
The binomial distribution with n Bernoulli trials and the probability of successes in each trial (p) is denoted as B(n, p) or β(n, p).

Example 1: State whether the following statement is true or false.
Tossing an unbiased coin 10 times is called a Bernoulli trial. 

Solution
The given statement is true because of the following reasons:
1. The coin is tossed 10 times, so the number of trials is 10, which is finite.
2. The trials are independent, as the outcome of one trial does not affect the outcomes of other trials.
3. There are two outcomes of tossing the coin, that is, a head and a tail.
4. As the trials are independent, the probability of success remains the same in each trial.

Example 2: An unbiased coin is tossed 10 times. What is the probability of getting 5 heads?

Solution
We have already seen that the trials of this experiment are Bernoulli trials.
Here,
Number of trials, n = 10
Probability of getting a head in the single trial, p = $\frac{1}{2}$
$\therefore q=1-p=1-\frac{1}{2}=\frac{1}{2}$
Let X be the random variable denoting the number of heads obtained in the 10 tosses of the coin. Then, X is a binomial distribution with parameters n = 10 and p = $\frac{1}{2}$ such that
$$\begin{gathered} P\left(X=x\right)={}^{10}C_x {\left(\frac{1}{2}\right)}^x {\left(\frac{1}{2}\right)}^{10- x}, x=0, 1, 2, ...,10 \\ \Rightarrow P\left(X=x\right)={}^{10}C_x {\left(\frac{1}{2}\right)}^{10} , x=0, 1, 2, ...,10 \end{gathered}$$
∴ Probability of getting 5 heads = $\mathrm{P}\left(X=5\right)={}^{10}C_5{\left(\frac{1}{2}\right)}^{10}=\frac{63}{256}$
Thus, the probability of getting 5 heads is $\frac{63}{256}$.

Example 3: Three dice are thrown simultaneously for 10 times. If getting a triplet is considered a success, then find the probability of trials that have less than 3 failures.

Solution
When three dice are thrown simultaneously, 216 outcomes (i.e. 6 × 6 × 6 outcomes) are obtained, out of which six are triplets. They are (1, 1, 1), (2, 2, 2…