What are the two requirements for a discrete probability distribution?
- Author: Noreen Niazi
- Last Updated on: December 16, 2023
A discrete probability distribution must satisfy two essential requirements:
- Each probability must be between 0 and 1, inclusive, and
- The sum of all probabilities must equal 1.
These conditions ensure that the distribution accounts for all possible outcomes and adheres to the fundamental principles of probability.
What are the two requirements for a discrete probability distribution?
A discrete probability distribution must satisfy two main requirements:
Probability values:
- Each probability assigned to an outcome must be a non-negative number.
- The sum of all probabilities for all possible outcomes must equal 1.
Mathematically, if X is a random variable representing the outcomes of an experiment, and x1, x2, x3, …, xn are the possible values of X, then the probability distribution must satisfy:
$$0 \leq P(X=x) \leq 1$$
$$\sum_{\text{all } x} P(X=x) = 1 $$
Mutual exclusivity:
- The events or outcomes must be mutually exclusive, meaning only one outcome can occur simultaneously. In other words, no two outcomes can occur simultaneously.
Example of Discrete Probability Distribution
For illustration, a roll of a six sided dice has results {1, 2, 3, 4, 5, & 6}, which are mutually exclusive so that only one number comes as an outcome of each roll.
Probabilities for any possible outcome must be legitimate, and their sum must be equal to one – to denote this certainty.
Practice questions
Here are some practice questions related to discrete probability distributions:
- Dice Roll: Imagine a balanced, six-faced die.w We consider X as a random variable representing an event such as a single roll. Determine for X an appropriate distribution.
- Coin Toss: Assume that you have a fair coin. Suppose, Y be a random variable denoting the number of heads in two successive throws. Define the probability distribution.
- Card Draw: There is a regular pack of fifty-two playing cards. A random variable is the number of red drawings made out of three consecutive draws without replacement. Define the probability distribution.
- Spinner Game: For example think of a spinner consisting of four equal segments called one, two, three, and four. Let w be a random variable describing the result of one spin. What is the probability distribution of W?
- Multiple Dice Rolls: For example, for if you toss two regular dices each with six sides. Considering that the random variable S refers to the total of obtained numbers. Give a distribution for S and its respective probability values.
Give a PMF or probability distribution table for each question that meets the specified conditions. Ensure that the probabilities range from zero to one, and that their sum equates to one.