In statistics and probability theory, a continuous random variable is a type of variable that can take any value within a given range. Unlike discrete random variables, which can only assume specific, separate values (like the number of students in a class), continuous random variables can assume any value within an interval, making them ideal for modelling quantities that vary smoothly without jumps.

This makes them ideal for modelling a wide range of real-world phenomena, such as the height of individuals, the time taken to complete a task, or the amount of rainfall in a particular period. In this article, we will discuss the concept of “Continuous Random Variable” in detail including its examples and properties. We will also discuss how it is different from a discrete random variable.

What is a Continuous Random Variable?

Continuous random variable is a type of random variable that can take on an infinite number of possible values within a given range. These values are typically real numbers, and the range can be either bounded or unbounded.

Unlike discrete random variables, which have countable outcomes, continuous random variables are associated with measurable and uncountable outcomes.

Examples of Continuous Random Variables

Continuous random variables can take any value within a given range and are commonly used in various fields to model and analyze real-world phenomena. Here are some examples:

• Height of Individuals: The height of people within a population can vary continuously. Measurements can be as precise as the measurement tool allows, such as 172.3 cm, 172.33 cm, etc.
• Weight of Objects: The weight of objects, such as fruits, animals, or packages, is another example. For instance, the weight of an apple can be 150.5 grams, 150.55 grams, and so on.
• Temperature: Temperature can be measured to a high degree of precision, such as 23.1°C, 23.12°C, and so forth. It is a continuous variable because it can take any value within the thermometric scale.
• Time: The time it takes to complete a task or event, like running a marathon, is a continuous random variable. For instance, a marathon might be completed in 3 hours, 2 minutes, and 47.5 seconds.
• Distance: The distance between two points can vary continuously. For example, the distance someone runs can be 5.123 kilometers, 5.1234 kilometers, etc.

Properties of Continuous Random Variables

Continuous random variables have several key properties that distinguish them from discrete random variables and are crucial for understanding their behavior and applications.

Probability Density Function (PDF)

A continuous random variable X is described by a probability density function f(x). The PDF f(x) gives the relative likelihood of X taking on a specific value x.

The PDF must satisfy two conditions: f(x) ≥ 0 for all  x: [Tex]\int_{-\infty}^{\infty} f(x) \, dx = 1[/Tex]

Cumulative Distribution Function (CDF)

The cumulative distribution function F(x) of a continuous random variable X represents the probability that X takes a value less than or equal to x:

[Tex]F(x) = P(X \leq x) = \int_{-\infty}^{x} f(t) \, dt[/Tex]

The CDF is non-decreasing and continuous, with: [Tex]\lim_{x \to -\infty} F(x) = 0 \quad \text{and} \quad \lim_{x \to \infty} F(x) = 1[/Tex].

Moment Generating Function (MGF)

The moment generating function of X, M_X(t), is defined as:

[Tex]M_X(t) = E(e^{tX}) = \int_{-\infty}^{\infty} e^{tx} \, f(x) \, dx[/Tex]

The MGF, if it exists, can be used to find all moments of X.

Characteristic Function

The characteristic function of X, ϕ_X(t), is the Fourier transform of the PDF:

[Tex]\phi_X(t) = E(e^{itX}) = \int_{-\infty}^{\infty} e^{itx} \, f(x) \, dx[/Tex]

It uniquely determines the distribution ofX.

Mean and Variance of Continuous Random Variable

Expectation (Mean): The expectation or mean of a continuous random variable X with PDF f(x) is given by:

[Tex]E(X) = \int_{-\infty}^{\infty} x \, f(x) \, dx[/Tex]

Variance: The variance of X measures the spread of the random variable around the mean and is given by:

[Tex]\text{Var}(X) = E[(X – E(X))^2] = \int_{-\infty}^{\infty} (x – \mu)^2 \, f(x) \, dx[/Tex]

Where E(X²) is the second moment of X: E(X²) = [Tex]\int_{-\infty}^{\infty} x^2 \, f(x) \, dx[/Tex]

Some other important properties are:

Probability of Intervals

• The probability that X lies within an interval [a, b] is given by the integral of the PDF over that interval: P(a ≤ X ≤ b) = [Tex] \int_{a}^{b} f(x) \, dx[/Tex]
• The probability of X taking any specific value x is zero, i.e., P(X = x) = 0.

Moments

Moments are quantitative measures related to the shape of the distribution. The n^th moment about the origin is given by:

[Tex]E(X^n) = \int_{-\infty}^{\infty} x^n \, f(x) \, dx[/Tex]

The n^th central moment (about the mean) is given by: [Tex]E[(X – \mu)^n] = \int_{-\infty}^{\infty} (x – \mu)^n \, f(x) \, dx[/Tex]

Skewness and Kurtosis

Skewness measures the asymmetry of the probability distribution of a real-valued random variable about its mean:

[Tex]\text{Skewness} = \frac{E[(X – \mu)^3]}{\sigma^3}[/Tex]

Kurtosis measures the “tailedness” of the probability distribution:

[Tex]\text{Kurtosis} = \frac{E[(X – \mu)^4]}{\sigma^4}[/Tex]

Common Continuous Random Variables

Some of the common distributions where continuous random variable is used are:

• Uniform Distribution
• Normal Distribution
• Exponential Distribution

Uniform Distribution

A continuous random variable X is uniformly distributed between a and b if its PDF is:

[Tex]f(x) = \begin{cases} \frac{1}{b – a} & \text{for } a \leq x \leq b \\ 0 & \text{otherwise} \end{cases}[/Tex]

Properties

• Each value in the interval [a, b] is equally likely.
• The mean is [Tex]\frac{a + b}{2}[/Tex].
• The variance is [Tex]\frac{(b – a)^2}{12}[/Tex].

Normal Distribution

A continuous random variable X is said to follow a normal distribution with mean μ and variance σ² if its PDF is given by:

[Tex]f(x) = \frac{1}{\sigma \sqrt{2\pi}} \exp\left( -\frac{(x – \mu)^2}{2\sigma^2} \right)[/Tex]

Properties

• Symmetrical about the mean μ.
• The mean, median, and mode are all equal.
• The distribution is described by the parameters μ (mean) and σ (standard deviation).

Exponential Distribution

A continuous random variable X follows an exponential distribution with rate parameter λ if its PDF is:

[Tex]f(x) = \begin{cases} \lambda e^{-\lambda x} & \text{for } x \geq 0 \\ 0 & \text{for } x < 0 \end{cases}[/Tex]

Properties

• Describes the time between events in a Poisson process.
• The mean is 1/λ.
• The variance is 1/λ².

Note: Other then these there are some distributions where continuous random variable is useful are:

• Gamma Distribution
• Beta Distribution
• Chi-Square Distribution
• Student’s t-Distribution

Continuous Random Variable vs Discrete Random Variable

The key differences between continuous and discrete random variables are listed in the following table: