The Cumulative Distribution Function and the Probability Density Function are two essential ideas in probability theory that frequently confound students. Understanding random variables’ behavior, features, and distributions depends critically on these operations. Knowing the differences between PDF and CDF is crucial to analyzing and interpreting the probabilities linked to continuous and discrete random variables. This article will discuss the definitions of cumulative distribution function vs probability density function and their unique roles and interactions. We will also offer a solved example to show the difference between PDF and CDF use.

The PDF is a crucial tool for understanding the **probabilities** associated with continuous random variables. It provides a smooth curve representing the probability distribution over possible values. The PDF function does not give the probabilities of specific individual values. Still, it describes the likelihood of the random variable taking on values within a small interval around a particular point.

To understand the concept of PDF, imagine a continuous probability distribution, such as the height of adult males. The probability for various height ranges will be displayed in the PDF. It might suggest, for instance, that people with heights between 5’9″ and 5’10” are more numerous than those with heights outside of that range.

The area under the PDF curve spanning a range represents the probability that the random variable will fall inside that range. To calculate the probability of a single value, which is the probability that the random variable will be infinitesimally close to that value, you must compute the integral of the PDF at that point.

When comparing cumulative distribution function vs probability density function, it’s essential to understand their distinct purposes and applications.

The CDF is a complementary concept to the **PDF** and provides a cumulative perspective of the probabilities associated with a random variable. Unlike the smooth curve of the PDF, the CDF is a step function that jumps at specific values. It displays the likelihood that a particular number will be less than or equal to the random variable.

The CDF begins at 0 for negative values, moving steadily towards 1 as the random variable’s value rises. For discrete random variables, the CDF increases in steps corresponding to the probabilities of each possible outcome. For continuous random variables, it increases smoothly, reflecting the accumulated probabilities across different intervals.

The CDF would demonstrate the likelihood of discovering a male with a height less than or equal to a certain value, such as 5 ‘9″, using the male heights example from before. The CDF allows us to respond to questions like “What percentage of adult males is shorter than 5 ‘9” by presenting cumulative probability.

Understanding how the Probability Density Function (PDF) vs Cumulative Distribution Function (CDF) interact is essential for comprehending how random variables behave and how their distributions work. Both functions provide complementary insights into the probabilities of the random variable’s values.

We previously showed how to compute the PDF vs CDF using the fair six-sided die example. Let’s now explore their connection and deeper aspects of their relationship.

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We need to integrate the PDF over a given range to find the CDF from the PDF. The CDF at a certain point x (F(x)) for a continuous random variable equals the region of the PDF curve up to that point. It can be modelled mathematically as follows:

F(x)=[a, x]f(t)dt

Here, x is the point on the distribution curve for which we wish to get the cumulative probability, and an is the lower limit of the range.

For our example of rolling the fair die, we can use the PDF values we previously calculated to find the CDF:

Let's calculate the CDF at x = 3: F(3) = ∫[1, 3] f(t) dt F(3) = ∫[1, 3] 16 dt F(3) = [t6] |[1, 3] F(3) = (36) - (16) F(3) = 26

Similarly, we can calculate the CDF for other values of x using the same approach.

The relationship between the PMF (Probability Mass Function) and the CDF is more apparent for discrete random variables. The PMF provides the probabilities for each specific value of the discrete random variable, while the CDF accumulates these probabilities.

The CDF at a particular value, x, is the sum of all the probabilities of the random variable being less than or equal to x. Mathematically, for discrete random variables:

F(x) = P(X ≤ x) = Σ[all values ≤ x] P(X = value)

By adding up the probabilities of all values up to x, we obtain the cumulative probability up to that point, which aligns with the CDF concept.

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Let us now understand the difference between PDF and CDF.

The CDF provides the probability that a random variable is less than or equal to a specific value, ‘x.’ The PDF represents the probability that the random variable takes on a precise value, ‘x.’

Let’s understand the unique properties and applications in PDF and CDP:

PDF | CDF |

The probability density function or PDF describes a continuous random variable’s probability distribution. It shows the probability that the random variable will have a particular value. | In general, the probability that a random variable will have a value less than or equal to a specific value is determined by the cumulative distribution function or CDF. |

PDF | CDF |

A continuous random variable is frequently represented using the expression f(x), where ‘x’ represents the variable’s value. | It can be applied to continuous and discrete random variables and is frequently expressed as F(x), where ‘x’ represents the variable’s value. |

PDF | CDF |

The PDF is used for continuous random variables, where the probability is distributed over an infinite range of values. | The CDF applies to discrete and continuous random variables, as it accumulates probabilities for all possible values of the random variable. |

PDF | CDF |

The PDF provides the probability density at a particular point on the continuous distribution curve, indicating how the probability is spread across different values. | The CDF gives the cumulative probability up to a specific value, offering insights into the probabilities of the random variable being less than or equal to that value. |

PDF | CDF |

The integral of the PDF over a certain range yields the probability of the random variable falling within that range. | The CDF is obtained by integrating the PDF from a lower bound to a specific value, ‘x’, which accumulates the probabilities up to that point. |

PDF | CDF |

The PDF can take any non-negative value for any given point on the distribution curve, representing the likelihood of the variable assuming that value. | The CDF always ranges from 0 to 1, as it gives the cumulative probability, and it is non-decreasing, meaning it can only increase or remain constant as ‘x’ increases. |

PDF | CDF |

The PDF is commonly used in probability density estimation, statistical modelling, and understanding the shape of continuous distributions. | The CDF can be used to determine a distribution’s percentiles and quantiles and the likelihood that a random variable will fall within a certain range. |

Understanding PDF and CDF differences is crucial for interpreting random variables’ distributions and behaviors in probability theory. The PDF and CDF serve distinct yet complementary roles: while the PDF provides the probability density of continuous random variables, showing the likelihood of values within specific intervals, the CDF accumulates probabilities, illustrating the likelihood of a variable being less than or equal to a particular value. Comparing the cumulative distribution function vs probability density function helps in appreciating their unique contributions to probability theory.

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A. The PDF and CDF are interrelated concepts in probability theory. The PDF gives the probability of a continuous random variable taking on a specific value. At the same time, the CDF provides the cumulative probability of the random variable being less than or equal to a given value.

A. The CDF and PDF are important in probability and statistics for describing random variable behavior. The CDF shows the cumulative probability up to a specific value “x” (denoted as “F(x)”). At the same time, the PDF displays the probability distribution of a continuous random variable (represented as “f(x)”).

A. PMF is for discrete random variables, giving probabilities for specific values. On the other hand, the PDF is for continuous random variables, showing the probability density over a range of values.

A. Both terms represent a mathematical function describing the probability distribution of a continuous random variable. Though “probability density function” and “probability distribution function” are interchangeable, they mean the same thing.

A. Yes, CDF is the integral of PDF for continuous variables. Think of it like this:

PDF: How likely a specific value is.

CDF: How likely a value less than or equal to that specific value is.

The CDF builds up the probability by integrating the PDF.