Understanding Continuous Distributions

Expert reviewed • 08 January 2025 • 16 minute read

understand and use the concepts of a probability density function of a continuous random variable
- know the two properties of a probability density function: $f(x)\geq0$ for all real $x$ and $\int_{-\infty}^\infty f(x)dx=1$
- define the probability as the area under the graph of the probability density function using the notation $P(X\leq r)=\int_a^r f(x)dx$ , where $f(x)$ is the probability density function defined on $[a,b]$
- examine simple types of continuous random variables and use them in appropriate contexts
- explore properties of a continuous random variable that is uniformly distributed
- find the mode from a given probability density function
obtain and analyse a cumulative distribution function with respect to a given probability density function
- understand the meaning of a cumulative distribution function with respect to a given probability density function
- use a cumulative distribution function to calculate the median and other percentiles

What are Continuous Distributions?

A continuous probability distribution is a way to determine the probability of all possible outcomes of a continuous random variable. A continuous random variable is a variable that can take any value within a given range. This differs from a discrete random variable, which can only take on distinct separate values.

The Difference Between Continuous and Discrete Distributions

Understanding the difference between continuous and discrete distributions can be difficult. In simple terms, a discrete distribution is similar to counting specific outcomes. For example, when counting coins, you can have a specific number of coins $(1 \;coin, 2\;coin, etc...)$ . Alternatively, a continuous distribution is like measuring a variable within a range. For example, when measuring a variable such as a length between the range of $1.5m-2m$ , there are infinite outcomes $(eg. \;1.60009m)$ .

Probability Density Functions

The Probability Density Function (PDF) of a continuous distribution describes the probability density at each point in the continuous range of the random variable. The probability that the variable falls within a certain interval is found by integrating the PDF over that interval. This is known as the cumulative distribution function, which will be explored later in the chapter.

In simple terms, a probability density function (PDF) describes the likelihood of a continuous random variable taking on a particular value. The PDF, denoted by $f(x)$ which lies within the bounds $[a,b]$ , has the following properties:

$f(x)\geq0$ for all $x$
The area under the entire PDF curve is $1$

∫_{a}^bf(x)dx=1

These properties will be vital to calculations related to the problems involving determining the cumulative distribution function over a given interval.

The Cumulative Distribution Function of a Continuous Distribution

As we now know, a continuous distribution operates with an infinite amount of outcomes within a certain interval. For the following formula, we use the interval $[a,b]$ . Thus, we let the a continuous random variable $X$ have values $(x)$ which lie in this interval. The cumulative distribution function (CDF) of this continuous distribution is denoted by $F(x)$ . As such, we can use the following formula:

As we now know, a continuous distribution operates with infinite outcomes over a given interval. For the following formula, we use the interval $[a,b]$ . Thus, we let a continuous random variable $X$ have values $(x)$ that lie in this interval. The cumulative distribution function (CDF) of this continuous distribution is denoted by $F(x)$ . As such, we can use the following formula:

F(x)=P(a\leq X\leq b)=\int_a^bf(x)dx

where,

$a$ and $b$ are scores that create an interval or range which can be replace with a value of $x$
$P(a\leq X\leq b)$ is the probability that $X$ falls within the interval $[a,b]$
$f(x)$ is the probability density function of the continuous distribution

As we can see from the formula above, the cumulative distribution function $F(x)$ is the integral of the PDF up to that value. Thus, another way to find the Probability of a score or interval can be to integrate the PDF. We can use the skills and knowledge gathered in module 4 to assist us with this task.

Additionally, the CDF is useful for calculating probabilities over intervals. For example, to find the probability that $X$ falls between two values $a$ and $b$ , you can use:

P(a<X≤b)=F(b)−F(a)

Specific Properties of the Cumulative Distribution Function

Let a continuous random variable $X$ **have values from a closed interval $[a,b]$ . The cumulative distribution function has the following properties:

$0≤F(x)≤1$ for all $x$ : This means that the probability of the function at any point in time can’t be negative. Additionally, the probability is not able to exceed $1$ .
As $x$ , which is the score approaches $a$ , $F(x)$ which is the probability of the function approaches $0$ :
As $x$ , which is the score approaches $b$ , $F(x)$ which is the probability of the function approaches

Practice Question 1

Imagine you have a light bulb that is expected to last on average 10 hours, and its lifespan is modelled with an exponential distribution. The exponential distribution’s PDF is defined by: $f(x)=0.1e^{0.1x}$ , for $x\geq0$ . Calculate the probability that the light bulb lasts between 9 and 11 hours.

Before we start answering the question, it is best to present all information in a neat order. The question is asking us to calculate the probability between 9 and 11 hours. Thus, we use the format for probability as discussed earlier in the chapter:

P(9≤X≤11)

Now, to find the probability, we must integrate the PDF, which is given in the question.

P(9≤X≤11)=\int_9^{11}0.1e^{0.1x}dx\\=[e^{0.1x}]_9^{11}\\=e^{1.1}-e^{0.9}\\\approx0.54

Uniform Continuous distributions

The uniform continuous distribution is a continuous probability distribution where all outcomes are equally likely within a certain range. This means that the continuous distribution is constant, represented graphically using a straight horizontal line for some value. The formula for the PDF and CDF alter slightly when we are dealing with uniform distributions.

The probability density function of a uniform distribution is constant between values $a$ and $b$ , and zero everywhere else. The formula for a PDF of a uniform continuous distribution is as follows:

f(x)=\frac{1}{b-a}

Here, the total area under the PDF curve is $1$ .

For the cumulative distribution function of a uniform distribution, values increases linearly from 0 to 1 as $x$ moves from $a$ to $b$ . The formula for the CDF is given by:

F(x)=\frac{x-a}{b-a}=\int_a^bf(x)dx

Practice Question 2

Determine the value for constant $m$ , that makes the function $f(x)=m$ , a probability density function (PDF), where $0\leq x\leq 8$ . Then find the corresponding cumulative distribution function (CDF)

From the knowledge presented in this chapter, we know that a PDF must be equal to $1$ . However, as this is a uniform continuous distribution, we can simply implement the formula provided above to determine the PDF.

f(x)=\frac{1}{b-a}\\m=\frac{1}{8-0}\\=\frac{1}{8}

Now, we can determine the CDF of this uniform distribution by integrating the value found for the PDF.

F(x)=\int_0^x\frac{1}{8}dt\\=[\frac{1}{8}t]_0^x\\=\frac{1}{8}x

Discrete & Continuous Probability Distributions: Means, Variances & CDFs

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Mean and Variance of the Normal Distribution

Return to Module 10: Continuous Probability Distributions