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Last updated on 8 September 2025

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Continuous Random Variable

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A continuous random variable is a random variable that can have a range of values within a specific range. It is used to represent measurements like weight, height, and time. In this article, we will explore the concepts and properties of continuous random variables in detail.

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What is a Continuous Random Variable?

A continuous random variable has an infinite number of possible values within a certain range. In statistics and probability theory, a continuous random variable plays a crucial role where the results are measured instead of being counted. A random variable is considered continuous if it can take any value within a specific interval. It is characterized by its possible values, which are infinite or uncountable. Also, using a function known as the probability density function, we can determine the probability that its value will fall inside a specific interval.

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What are the Properties of Continuous Random Variables?

To understand the probability distributions and conduct statistical analysis, we need to comprehend the various properties of a continuous random variable. Several important properties distinguish the continuous random variable from the discrete random variable. They include: 

 

Probability Density Function

 

A probability density function f(x) defines a continuous random variable x. The relative probability for x to be a certain value x is represented by the PDF f(x). The PDF needs to meet two conditions. The first one is f(x) ≥ 0 for all  x. It means the probability density function (PDF) must always be non-negative but can be zero for some values. The next condition is that the total probability over all possible values of x must be 1, meaning the sum of all probability values must be 100% or 1. This is written as:
  

\(\int_{-\infty}^{\infty} f(x)\, dx = 1 \)

 

Cumulative Distribution Function (CDF)

 

The cumulative distribution function (CDF), represented as F(x), shows the likelihood that a continuous random variable X will take a value that is less than or equal to x: 

 

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

 

The CDF is non-decreasing and continuous. The CDF approaches 0 as x moves closer to negative infinity.
lim ​F(x) = 0
x→∞

 

Likewise, the CDF approaches 1 as x moves closer to positive infinity.  
lim ​F(x) = 1
x→∞ 

 

Moment Generating Function (MGF)

 

The moment generating function (MGF) of a continuous random variable X is represented as Mx (t). It is defined as:


\(M_X(t) = E\!\left(e^{tX}\right) = \int_{-\infty}^{\infty} e^{tx} f(x)\,dx \)

 

The MGF can be used to find all the moments of X, including its mean and variance if it exists.

 

Characteristic Function

 

The characteristic function of a continuous random variable X is expressed as ϕX(t). It is the Fourier transform of the probability density function (PDF). 


\(\varphi_X(t) = E\!\left(e^{itX}\right) = \int_{-\infty}^{\infty} e^{itx} f(x)\,dx\)

 

It determines the distribution of X uniquely.

 

Mean and Variance of Continuous Random Variable

 

A continuous random variable X with PDF f(x) has the following expectation or mean:

 

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

 

The formula represents the expected value of X. 

 

The variance of X measures the average of the squared deviations of the random variable from the mean. It is defined as:

 

\(\operatorname{Var}(X) = E\!\left[(X - E(X))^{2}\right] = \int_{-\infty}^{\infty} (x - \mu)^{2} f(x)\,dx \)

 

Here, μ = E (X) is the mean

 

Where X's second moment is represented by E(X2) and it is denoted as:

 

\(E(X^2) = \int_{-\infty}^{\infty} x^2 f(x)\,dx \)

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Continuous Random Variable Formulas

The probabilities associated with a continuous random variable are described by the probability density function (PDF) and cumulative probability function (CDF). Here are some key formulas related to continuous random variables. 

 

PDF of Continuous Random Variable

 

PDF of a continuous random variable is a function that shows the likelihood of the variable taking on different values. It does not provide probabilities for specific values but rather for intervals. X is the continuous random variable, and the formula for the PDF, f(x), is:

 

f(x) = dF(x) / dx = F'(x)

 

Here, F(x) is the cumulative distribution function. 

 

CDF of Continuous Random Variable


It describes the likelihood that a specific value, x, will be equal to or less than the random variable, X. CDF of a continuous random variable can be found by integrating the PDF. The formula is calculated between two points, a and b. The formula for CDF of continuous random variable is: 

 

\(P(a < X \leq b) = F(b) - F(a) = \int_{a}^{b} f(x)\,dx \)

 

The Mean of Continuous Random Variable

 

The mean of the continuous random variable, X, is its expected value. It is calculated as a weighted average of all possible values. Also, each value is weighted according to the probability density function (PDF). The formula is:

 

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

 

The Variance of Continuous Random Variable

 

It measures how much the values of X deviate from the mean. It is the expected value of the squared differences between the variable and its mean. The formula is given as follows:

 

\(\operatorname{Var}(X) = \sigma^{2} = \int_{-\infty}^{\infty} (x - \mu)^{2} f(x)\,dx \)

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Difference Between Discrete Random Variable And Continuous Random Variable

A random experiment’s numerical outcome is called a random variable. The two types of random variables are discrete random variable and continuous random variable. Let us look at the differences between them:

 

Continuous Random Variable

 

Discrete Random Variable

 

Can have any value within a specified range

 

Can only have particular and separate values

 

The possible values are infinite within a certain range. Example: all real values between 1 and 2

 

The possible values are finite or countably infinite. Example: 1, 2, 3, and so on.

 

A probability density function (PDF) describes a continuous random variable

 

A probability mass function describes a discrete random variable

 

The probability of a single value is zero (P (X = x) = 0

 

The probability of a single value is non-zero (P (X = x) > 0

 

It is represented by a smooth curve

 

It is represented by bar graphs

 

Examples include time, distance, temperature, height, and weight

 

Examples include the number of children, the number of flowers, and the results of a die roll

 

 

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What are the Types of Continuous Random Variable?

We use continuous random variables to model situations that involve measurements. For example, if we want to know the possible amount of rainfall over a year or temperature on any given day. Following are the significant continuous random variables linked to certain probability distributions: 

 

Uniform Random Variable

 

A uniform random variable represents a uniform distribution, which describes events with equal chances of happening. A uniform random variable’s PDF is as follows: 

 

Otherwise, a ≤ x ≤ b 
Here, a and b are the lower and upper bounds of the distribution. 
 

Normal Random Variable

 

A normal random variable is a continuous random variable that is used to model a normal distribution. If a normal distribution’s parameters are expressed as X ~ N (μ, σ2), then the following is the formula for the PDF:

 

Here, μ is the mean 
σ is the standard deviation 
σ2 is the variance
 

Exponential Random Variable

 

An exponential distribution is a continuous probability distribution used to model processes in which a specific number of events occur continuously and independently at a constant average rate λ, where λ ≥ 0. The exponential random variable follows an exponential distribution. The following is the PDF of an exponential random variable:

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Real-life Applications of Continuous Random Variable

Within a specified range, a continuous random variable can have an infinite number of values. This concept is widely used in situations where measurements are involved. Here are some real-world significance of continuous random variables. 

 

  • Weather forecasters employ the continuous random variable to model temperature, wind speed, rainfall, and humidity. For instance, the probability of rainfall within a certain range is calculated by using normal distribution. 

 

  • The values that fluctuate continuously like stock prices, interest rates, and exchange rates can be modeled by log-normal distribution. 

 

  • To assess the variations in production and manufacturing, we can use the Gaussian distribution. Also, the variables like weight, length, or thickness of products follow a continuous distribution.

 

  • Continuous variables in transportation, including travel time, vehicle speed, and fuel consumption, are used in urban planning and traffic management. 
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Common Mistakes and How to Avoid Them in Continuous Random Variable

A continuous random variable is a random variable that can take any value within a specific range. Knowing the key properties and concepts of this variable helps us to solve various mathematical problems, and it improves our problem-solving skills. Here are some of the common mistakes and their helpful solutions to avoid these common errors. 

Mistake 1

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Assuming a continuous random variable can take exact values

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Keep in mind that there is no chance of one precise value for continuous variables. Only probabilities over a range of values are found. For instance, the probability of a student’s height being exactly 160 cm is 0, but the probability of height between 160 cm and 165cm is a real value.

Mistake 2

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Believing that probabilities are merely numerical values

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Students should remember that the region under the probability density function (PDF) curve between two points is where probabilities originate. They should remember that probabilities are areas under the curve.

Mistake 3

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Confusion between PDF and CDF

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Learn each concept thoroughly. PDF indicates how likely a value is, and it doesn’t provide a straight probability. The likelihood of obtaining a value that is less than or equal to a certain number is provided by CDF. For example, if the CDF at 10 is 0.7, it means there is a 70% chance the value will be less than or equal to 10.

Mistake 4

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Forgetting to check the total probability equals 1

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Students should check the total probability equals 1 or 100%. Otherwise, it will lead to incorrect conclusions. If the total probability is not 1, then there might be a mistake in the calculation.

Mistake 5

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Neglecting the value of mean

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Don’t forget that the mean is the average of numerous trials. It is not the value that occurs the most.  Sometimes students think that the mean is the value that occurs most in a distribution.

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Solved Examples of Continuous Random Variable

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Problem 1

A random variable X has the probability density function (PDF): f (x) = k(5−x), 0 ≤ x ≤5. Find the value of k.

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0.08

Explanation

A function must adhere to the following basic rule in order to be considered a valid probability density function (PDF): 

 

\(\int_{-\infty}^{\infty} f(x)\,dx = 1 \)

 

As we know, the given PDF is defined for 0 ≤ x ≤5, then the integral is taken over this range:
 

\(\int_{0}^{5} f(x)\,dx = 1 \)

 

Next, we can substitute f(x) = k (5 - x): 

 

So, now we have to factor out the constant k:

 

\(k \int_{0}^{5} (5 - x)\,dx = 1 \)

 

Then, we have to integrate (5 - x) term by term: 

∫ (5 - x) dx = ∫ 5 dx - ∫ x dx 

 

Here, we have to use the basic integration rules:

∫ 5 dx = 5x
∫ x dx = x2 / 2

 

So the antiderivative is:
 

\(\int (5 - x)\,dx = 5x - \frac{x^2}{2} \)

 

Next, we evaluate the definite integral from 0 to 5: 
 

\(\begin{align*} \int_{0}^{5} (5 - x)\,dx &= \left[ 5x - \frac{x^2}{2} \right]_0^5 \\ &= \left( 5(5) - \frac{5^2}{2} \right) - \left( 5(0) - \frac{0^2}{2} \right) \\ &= (25 - 12.5) - 0 \\ &= 12.5 \end{align*} \)

 

Substitute back to solve for k:

So, k (12.5) = 1
k = 1 / 12.5 
k = 0.08 

 

Hence, the value of k is 0.08.  

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Problem 2

For a continuous random variable with PDF: f (x)=4x2, 0 ≤ x ≤ 1. Find the value of E (X).

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1

Explanation

Given: f (x) = 4x2, 0 \(\le \) x \(\le \) 1

 

We have to find E (X) of X. Here, we can use the formula:

 

\(E(X) = \int_{a}^{b} x f(x)\,dx\)

 

Where a = 0 and b = 1

 

Now, we have to substitute f (x)

 

\(E(X) = \int_{0}^{1} x (4x^2)\,dx = \int_{0}^{1} 4x^3\,dx\)

 

Integrating,

 

\(\int 4x^3\,dx = x^4 + C\)

 

Evaluating the definite integral from 0 to 1,

 

\(E(X) = \left[ x^4 \right]_0^1 = 1^4 - 0^4 = 1 \)= 1

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Problem 3

Find the mean E(X) for the uniform distribution U (2,10).

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6

Explanation

The formula for the mean of a uniform distribution U (a, b) is:

 E (X) = (a + b) / 2  

From U (2,10), we have a = 2 and b = 10, so:

 E (X) = (2 + 10) / 2

= 12 / 2 = 6

Hence, the mean is 6.

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Problem 4

Find the median m such that P (X ≤ m) = 0.6 for the uniform distribution U (0, 5).

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3

Explanation

Here, to find the value of the median, we have to use the formula:

F(m) = (m - a) / (b - a) 

From U (0,5), this becomes: 

F(m) = (m - 0) / 5 = m / 5 

Setting F(m) = 0.6 

m / 5 = 0.6 

Next, we can solve for m:

m = 0.6 × 5 

m = 3

The median is 3.

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Problem 5

Find the median m such that P (X ≤ m) = 0.5 for the uniform distribution U (2, 8).

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5

Explanation

F(m) = (m - a) / (b - a) 

From U (2,8), this becomes: 

F(m) = (m - 2) / (8 - 2) = (m - 2) / 6

Setting F(m) = 0.5:

(m - 2) / 6 = 0.5 

Next, we can solve for m:

m - 2 = 0.5 × 6

m - 2 = 3 

m = 5 

So, the median is 5. 

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FAQs on Continuous Random Variables

1.Define a continuous random variable.

A continuous random variable is a type of random variable that can take any value within a specific interval. It is used to represent quantities that are measured rather than counted, like weight, height, time, and temperature.

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2.Differentiate a continuous random variable and a discrete random variable.

A discrete random variable has a countable set of possible values, such as rolling a die. While a continuous random variable can take an unlimited number of values within an interval. 

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3.What do you mean by probability density function (PDF)?

The distribution of values of a continuous random variable is described by a probability density function (PDF). A probability density function f(x) defines a continuous random variable X. The relative probability for X to be a certain value x is represented by the PDF f(x). 

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4.Define cumulative distribution function (CDF).

The cumulative distribution function (CDF), represented as F(x), shows the likelihood that a continuous random variable X will take a value that is less than or equal to x. It is found by integrating the PDF up to that specific value.  

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Jaipreet Kour Wazir

About the Author

Jaipreet Kour Wazir is a data wizard with over 5 years of expertise in simplifying complex data concepts. From crunching numbers to crafting insightful visualizations, she turns raw data into compelling stories. Her journey from analytics to education ref

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