Probability Distributions

Understanding the basics of probability distributions

A probability distribution is a mathematical function used to illustrate the likelihood of an outcome or an event. In other words, it is a function used illustrate the possible values that a variable can take randomly. For instance, if you take a group of people randomly and then measure their weight in kilograms, it is possible to cerate a distribution of their weight. This created distribution can help one determine which outcomes are more likely than others and the likelihood that a different result might be observed. The following is a summary of the general properties of all probability distributions;

• Probability distributions can simply be defined as a function to illustrate the likelihood of a certain outcome or event. The following notation is widely used and accepted; P(x)= the probability or likelihood that a certain random variable will take the value of x
• The sum of all likelihood values/ probabilities in a certain event must be equal to 1 (one). In this case, all likelihoods values/ probabilities fall between 0 and 1 and can therefore be expressed as percentage between 0% and 100%. The closer a probability of an event or outcome is to 1, the higher the chance of it occurring.
• There exist different kinds of probability distributions determined by the type of variables in question. These are;
  • Discrete probability distributions used to illustrate likelihood of events in discrete variables.
  • Probability density functions (PDF) distributions used to illustrate likelihood of events and outcomes in scale/ continuous variables.

These distributions also referred to as probability mass functions. Discrete probability functions can only assume individually separate and distinct values (Discreet values). Into perspective, rolling a 6-sided dice, tossing a coin or counting the number of a certain outcome are individually separate and distinct values. For example, when you toss a fair sided coin with heads and tails, you can only have the outcome, head or tails, there is no value that is in between heads and tails. In another example, if you roll 6-sided fair dice, you can only have the outcomes, 1, 2, 3, 4, 5 or 6, there are no values that will come in between any of these events. The following is a summary of the general properties of a discrete probability distribution;

• In a discrete probability distribution, the likelihood of any given event or outcome can never be zero. On the other hand, the likelihoods of all possible events or outcomes will and should always sum up to 1 (one). For example, when you roll a 6-sided fair dice, there will be 6 discrete outcomes with each outcome having a 1/6 probability/ likelihood of occurring. The total sum of the probability will be 6/6 which is also equal to 1.
• Based on the type of data that is being examined, different types of discrete probability distributions are applied. The correct type t use will depend of the properties of the variables in question. These distribution functions are;
  1. Binomial distribution functions. This function is used to illustrate the likelihood of events in binary data such as the toss of a coin which will have only two outcomes.
  2. Poisson distributions functions. This function is used to illustrate the likelihood of events in count variables such as the number of cars sold in an hour.
  3. Uniform distribution function. This function illustrates the likelihood of events that have the same probability such as rolling a 6-sided fair dice.

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