Probability Distribution

A probability distribution is a function that gives the individual probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events.

Expressing a probability distribution

A probability distribution is often described in the form of an equation or a table that links each outcome of a probability experiment with its corresponding probability of occurring.

Example of expressing probability distribution 1

Consider an experiment where the random variable X = the score when a fair dice is rolled.

Since there are six equally likely outcomes here, the probability of each outcome is \(\frac{1}{6}\).

Solution 1

The corresponding probability distribution can be described:

• As a probability mass function:

\(P (X = x) = \frac{1}{6}\), x = 1, 2, 3, 4, 5, 6

• In the form of a table:

Example of expressing probability distribution 2

A fair coin is tossed twice in a row. X is defined as the number of heads obtained. Write down all the possible outcomes, and express the probability distribution as a table and as a probability mass function.

Solution 2

With heads as H and tails as T, there are 4 possible outcomes:

(T, T), (H, T), (T, H) and (H, H).

Therefore the probability of getting \((X = x = \text{number of heads} = 0) = \frac{\text{number of outcomes with 0 heads}} {\text{total number of outcomes}} = \frac{1}{4}\)

\((x = 1) = \frac{\text{number of outcomes with 1 heads}} {\text{total number of outcomes}} = \frac{2}{4}\)

\((x = 2) = \frac{\text{number of outcomes with 2 heads}} {\text{total number of outcomes}} = \frac{1}{4}\)

Now let's express the probability distribution

• As a probability mass function:

\(P (X = x) = 0.25, \space x = 0, 2 = 0.5, \space x = 1\)

• In the form of a table:

Example of expressing probability distribution 3

The random variable X has a probability distribution function

\(P (X = x) = kx, \space x = 1, 2, 3, 4, 5\)

What is the value of k?

Solution 3

We know that the sum of the probabilities of the probability distribution function has to be 1.

For x = 1, kx = k.

For x = 2, kx = 2k.

And so on.

Thus, we have \(k + 2k + 3k + 4k + 5k = 1 \Rightarrow k = \frac{1}{15}\)

Discrete and continuous probability distribution

Discrete probability distribution function

Mathematically, a discrete probability distribution function can be defined as a function p (x) that satisfies the following properties:

1. The probability that x can take a specific value is p (x). That is \(P (X = x) = p (x) = px\)
2. p (x) is non-negative for all real x.
3. The sum of p (x) over all possible values of x is 1, that is \(\sum_jp_j = 1\)

A discrete probability distribution function can take a discrete set of values – they need not necessarily be finite. The examples we have looked at so far are all discrete probability functions. This is because the instances of the function are all discrete – for example, the number of heads obtained in a number of coin tosses. This will always be 0 or 1 or 2 or… You will never have (say) 1.25685246 heads and that is not part of the domain of that function. Since the function is meant to cover all possible outcomes of the random variable, the sum of the probabilities must always be 1.

Further examples of discrete probability distributions are:

• X = the number of goals scored by a football team in a given match.

• X = the number of students who passed the mathematics exam.

• X = the number of people born in the UK in a single day.

Discrete probability distribution functions are referred to as probability mass functions.

Continuous probability distribution function

Mathematically, a continuous probability distribution function can be defined as a function f (x) that satisfies the following properties:

1. The probability that x is between two points a and b is \(p (a \leq x \leq b) = \int^b_a {f(x) dx}\)
2. It is non-negative for all real x.
3. The integral of the probability function is one that is \(\int^{-\infty}_{\infty} f(x) dx = 1\)

A continuous probability distribution function can take an infinite set of values over a continuous interval. Probabilities are also measured over intervals, and not at a given point. Thus, the area under the curve between two distinct points defines the probability for that interval. The property that the integral must be equal to one is equivalent to the property for discrete distributions that the sum of all the probabilities must be equal to one.

Examples of continuous probability distributions are:

• X = the amount of rainfall in inches in London for the month of March.
• X = the lifespan of a given human being.
• X = the height of a random adult human being.

Continuous probability distribution functions are referred to as probability density functions.

Cumulative probability distribution

A cumulative probability distribution function for a random variable X gives you the sum of all the individual probabilities up to and including the point x for the calculation for P (X ≤ x).

This implies that the cumulative probability function helps us to find the probability that the outcome of a random variable lies within and up to a specified range.

Example of cumulative probability distribution 1

Let's consider the experiment where the random variable X = the number of heads obtained when a fair dice is rolled twice.

Solution 1

The cumulative probability distribution would be the following:

The cumulative probability distribution gives us the probability that the number of heads obtained is less than or equal to x. So if we want to answer the question, “what is the probability that I will not get more than heads”, the cumulative probability function tells us that the answer to that is 0.75.

Example of cumulative probability distribution 2

A fair coin is tossed three times in a row. A random variable X is defined as the number of heads obtained. Represent the cumulative probability distribution using a table.

Solution 2

Representing obtaining heads as H and tails as T, there are 8 possible outcomes:

(T, T, T), (H, T, T), (T, H, T), (T, T, H), (H, H, T), (H, T, H), (T, H, H) and (H, H, H).

The cumulative probability distribution is expressed in the following table.

Example of cumulative probability distribution 3

Using the cumulative probability distribution table obtained above, answer the following question.

1. What is the probability of getting no more than 1 head?

2. What is the probability of getting at least 1 head?

Solution 3

1. The cumulative probability P (X ≤ x) represents the probability of getting at most x heads. Therefore, the probability of getting no more than 1 head is P (X ≤ 1) = 0.5
2. The probability of getting at least 1 head is \(1 - P (X ≤ 0) = 1 - 0.125 = 0.875\)

Uniform probability distribution

A probability distribution where all of the possible outcomes occur with equal probability is known as a uniform probability distribution.

Thus, in a uniform distribution, if you know the number of possible outcomes is n probability, the probability of each outcome occurring is \(\frac{1}{n}\).

Example of uniform probability distribution 1

Let us get back to the experiment where the random variable X = the score when a fair dice is rolled.

Solution 1

We know that the probability of each possible outcome is the same in this scenario, and the number of possible outcomes is 6.

Thus, the probability of each outcome is \(\frac{1}{6}\).

The probability mass function will therefore be, \(P (X = x) = \frac{1}{6}, \space x = 1, 2, 3, 4, 5, 6\)

Binomial probability distribution

Binomial Distribution is a probability distribution function that is used when there are exactly two mutually exclusive possible outcomes of a trial. The outcomes are classified as "success" and "failure", and the binomial distribution is used to obtain the probability of observing x successes in n trials.

Intuitively, it follows that in the case of a binomial distribution, the random variable X can be defined to be the number of successes obtained in the trials.

You can model X with a binomial distribution, B (n, p), if:

• there are a fixed number of trials, n

• there are 2 possible outcomes, success and failure

• there is a fixed probability of success, p, for all trials

• the trials are independent