A probability distribution provides a comprehensive description of how probabilities are allocated among the possible values of a random variable. In statistics and probability theory, a random variable represents a numerical outcome determined by a random phenomenon. The probability distribution, therefore, serves as a mathematical model for expressing the uncertainty associated with such outcomes.
Random variable definition:
If you flip a coin, there are two possible outcomes: heads or tails. Assuming it’s a fair coin, the probability of getting heads is 50%, and the probability of getting tails is also 50%. Each flip is an independent event, meaning the outcome of one flip doesn’t affect the outcome of any other flip. If you flip the coin twice there two possibilities.
In probability, a random variable is a real valued function whose domain is the sample space of the random experiment. It means that each outcome of a random experiment is associated with a single real number, and the single real number may vary with the different outcomes of a random experiment. Hence, it is called a random variable and it is generally represented by the letter “X”.
For example, let us consider an experiment for tossing a coin two times.
Hence, the sample space for this experiment is S = {HH, HT, TH, TT}, [H represents Heads, and T represents Tails when you flip the coin]
If X is a random variable and it denotes the number of heads obtained, then the values are represented as follows:
X(HH) = 2, X(HT) = 1, X(TH) = 1, X(TT) = 0.
Similarly, we can define the number of tails obtained using another variable, say Y.
(i.e) Y(HH) = 0, Y(HT) = 1, Y(TH) = 1, Y(TT)= 2.
Key Concepts
- Random Variable: A variable whose values result from the outcomes of a random experiment. These values are numerical in nature.
- Probability Distribution: A function that assigns a probability to each potential value that a random variable can assume. It characterizes the behavior and likelihood of different outcomes.
- Discrete Random Variable: A random variable that can take on a finite or countably infinite set of values. For example, the number of heads obtained when flipping a coin three times.
- Continuous Random Variable: A variable that can take any value within a given interval. An example would be the height of an individual.
Types of Probability Distributions
- Probability Mass Function (PMF): Used for discrete random variables, the PMF specifies the probability associated with each distinct value.
- Probability Density Function (PDF): Applicable to continuous random variables, the PDF indicates the relative likelihood of the variable assuming a particular value within a range.
- Cumulative Distribution Function (CDF): Relevant to both discrete and continuous variables, the CDF provides the probability that the random variable is less than or equal to a specified value.
Illustrative Example
Consider an experiment in which a fair coin is flipped three times. Let the random variable X represent the number of heads observed. The possible values of X are 0, 1, 2, and 3, with the corresponding probabilities as follows:
The probabilities are:
P(X=0) = 1/8 (all tails)
P(X=1) = 3/8 (one head, two tails)
P(X=2) = 3/8 (two heads, one tail)
P(X=3) = 1/8 (all heads)
This set of probabilities, along with the possible values of X, forms the probability distribution of the random variable X.
This tabulation of possible outcomes and their probabilities constitutes the probability distribution of the random variable X.
Significance of Probability Distributions
Probability distributions are fundamental to the study of probability and statistics. They enable us to:
- Model and analyze random phenomena
- Predict potential outcomes
- Determine the likelihood of events
- Compare different random variables
- Conduct statistical inference and hypothesis testing
Understanding probability distributions is essential for making data-driven decisions and interpreting statistical results in various domains, including science, economics, engineering, and social sciences.
