In the realm of probability and statistics, a random variable is a variable whose possible values are numerical outcomes of a random phenomenon. Random variables are broadly classified into two types: discrete and continuous.
A discrete random variable can only take on a countable number of distinct values. For example, the number of heads in a series of coin flips or the outcome of a dice roll are discrete variables.
In contrast, a continuous random variable can take on an infinite number of possible values within a given range. These variables are typically associated with measurements, such as the height of a person, the temperature of a room, or the time it takes to complete a task. Because a continuous variable can take on any value within an interval, the probability of it being exactly equal to a specific value is zero. Instead, we focus on the probability of the variable falling within a certain range.
The behavior of a continuous random variable is described by its Probability Density Function (PDF), denoted as $f(x)$. The PDF does not give the probability of a specific value directly; rather, it describes the relative likelihood of a random variable taking on a value within a particular interval. The probability is represented by the area under the curve of the PDF over that interval.
A valid PDF must satisfy two key properties:
The PDF gives the shape of the distribution, and its peak indicates where the values are most likely to occur.
While the PDF describes the probability density, the Cumulative Distribution Function (CDF), denoted as $F(x)$, gives the probability that a random variable $X$ will take a value less than or equal to a specific value $x$. It is defined as:
$$ F(x) = P(X \leq x) = \int_{-\infty}^{x} f(t) \,dt $$The CDF accumulates probability as the value of $x$ increases. Consequently, it is a non-decreasing function with a starting value of 0 and a maximum value of 1. The CDF is particularly useful for finding the probability that a variable falls within a specific range, say between $a$ and $b$, which can be calculated as $P(a < X \leq b) = F(b) - F(a)$.
Several standard distributions are frequently used to model real-world phenomena. Here, we'll explore three of the most common: the uniform, exponential, and normal distributions.
The uniform distribution, sometimes called the rectangular distribution, is characterized by a constant probability density over a specified interval $[a, b]$. This means that all outcomes within this range are equally likely. It's often used in random number generation.
For example, if a bus arrives every 20 minutes, and you arrive at the bus stop at a random time, your waiting time would follow a uniform distribution between 0 and 20 minutes. You can explore this distribution with the calculator below.
The exponential distribution is used to model the time until an event occurs in a Poisson process, where events happen independently and at a constant average rate. Examples include the lifespan of an electronic component or the time between customer arrivals at a service desk.
A key feature of the exponential distribution is its memoryless property. This means that the probability of an event occurring in the future is independent of how much time has already passed. For instance, if a light bulb's lifespan follows an exponential distribution, the probability that it will last for another 100 hours is the same whether it has been working for 10 hours or 1000 hours.
Experiment with the exponential distribution using the interactive calculator below.
The normal distribution, or Gaussian distribution, is arguably the most important distribution in statistics. It is characterized by its symmetric, bell-shaped curve. Many natural phenomena, such as height, weight, and blood pressure, tend to follow a normal distribution.
It is defined by two parameters: the mean ($\mu$), which determines the center of the distribution, and the standard deviation ($\sigma$), which controls its spread.
The CDF of the normal distribution does not have a simple closed-form expression and is typically found using statistical tables or software.
Continuous random variables are essential for modeling and understanding a wide array of real-world phenomena that involve measurements. From analyzing stock market returns to controlling the quality of manufactured products, continuous distributions provide a powerful framework for quantifying uncertainty and making informed decisions. By understanding the concepts of PDFs and CDFs, and becoming familiar with common distributions like the uniform, exponential, and normal, one gains a foundational toolkit for tackling complex problems in science, engineering, and beyond.
A Probability Density Function (PDF) describes the relative likelihood of a continuous random variable taking on a specific value, where the probability is the area under the curve. A Cumulative Distribution Function (CDF) gives the probability that the variable will be less than or equal to a certain value. The CDF is the integral of the PDF.
Because a continuous random variable can take on an infinite number of values within any given interval, the likelihood of it landing on any single, exact point is infinitesimally small, and thus considered to be zero. Probability for continuous variables is only meaningful over intervals.
The memoryless property signifies that the future probability of an event occurring is not dependent on its past history. For example, if the time between arrivals at a store is exponentially distributed, the probability of a customer arriving in the next five minutes is the same regardless of how long it has been since the last customer arrived.