Support

Definition

Support is the force or structure that holds up or maintains something, providing stability, strength, and assistance. In the context of continuous probability density functions, support refers to the range of values over which the function is defined and non-zero.

5 Must Know Facts For Your Next Test

1. The support of a continuous probability density function is the interval or range of values over which the function is defined and non-zero.
2. The support of a PDF is often denoted as the interval $[a, b]$, where $a$ and $b$ are the lower and upper bounds of the support, respectively.
3. The support of a PDF must be a finite, contiguous interval, and the function must be non-negative over this interval.
4. The total area under the curve of a PDF over its support must be equal to 1, representing the total probability of the random variable taking on a value within the support.
5. The support of a PDF is an important characteristic that helps define the range of possible values for the random variable and the shape of the probability distribution.

Review Questions

• Explain the significance of the support of a continuous probability density function.
  • The support of a continuous probability density function (PDF) is the interval or range of values over which the function is defined and non-zero. This is a crucial characteristic because it determines the set of possible values that the random variable can take on, and it directly affects the shape and properties of the probability distribution. The support must be a finite, contiguous interval, and the total area under the curve of the PDF over its support must be equal to 1, representing the total probability of the random variable taking on a value within the support.
• Describe how the support of a PDF is related to the domain of the function.
  • The support of a continuous probability density function (PDF) is closely related to the domain of the function. The domain of a PDF is the set of all possible values that the random variable can take on, while the support is the specific interval or range of values over which the function is defined and non-zero. The support is a subset of the domain, as the PDF must be non-negative and have a finite, contiguous interval as its support. The support is a critical characteristic that helps define the range of possible values for the random variable and the shape of the probability distribution.
• Analyze the relationship between the support of a PDF and the total probability of the random variable taking on a value within that support.
  • The support of a continuous probability density function (PDF) is directly related to the total probability of the random variable taking on a value within that support. By definition, the total area under the curve of the PDF over its support must be equal to 1, representing the total probability of the random variable taking on a value within the support. This means that the support of the PDF defines the range of possible values for the random variable, and the shape of the PDF within this support determines the relative likelihood of the random variable taking on different values within that range. Understanding the support of a PDF is crucial for analyzing and interpreting the probability distribution of a continuous random variable.

Related terms

Probability Density Function (PDF): A function that describes the relative likelihood of a random variable taking on a given value in a continuous probability distribution.

Continuous Random Variable: A random variable that can take on any value within a given interval or range, rather than just discrete values.

Domain: The set of all possible values that a function can take on.