The mean of a probability density function (PDF) is found by calculating the expected value, which involves integrating the product of the variable and the PDF over its entire range.
Understanding the Mean of a PDF
The mean, often denoted by μ (mu), represents the average value of a random variable described by a PDF. It is also known as the expected value. Unlike the mean of a set of discrete numbers, the mean of a PDF requires integration because the variable can take on a continuous range of values.
The Formula
The formula to find the mean (expected value, E[X]) of a continuous random variable with a PDF denoted as f(x) is:
-
E[X] = ∫x * f(x) dx
- Where:
- ∫ represents the integral sign.
- x is the random variable.
- f(x) is the probability density function.
- The integration is carried out over the entire range of possible values for x.
- Where:
Steps to Calculate the Mean
To find the mean of a PDF, follow these steps:
- Identify the PDF: Determine the specific probability density function, f(x), that you are working with.
- Determine the range: Identify the domain over which the PDF is defined. This is the integration range.
- Multiply x by f(x): Create a new function by multiplying the variable x by the PDF, f(x).
- Integrate: Perform the integration of x f(x) with respect to x* over the defined range of the PDF. This result is the expected value, or mean.
Example (Based on Reference)
While the reference video does not provide the explicit formula or steps, it describes solving for the mean through integration. It mentions the result of such a calculation being 1.8484, without showing the explicit function. This demonstrates that the process involves setting up an appropriate integral and evaluating it to obtain a numerical value for the mean.
- Reference insight: The video mentions the integration result of 1.8484, which is a numerical mean calculated by setting up and solving the correct integral (as described in the formula).
Summary
| Step | Description |
|---|---|
| 1. Identify the PDF | Define the probability density function f(x). |
| 2. Integration Range | Determine the valid range of the probability density function . |
| 3. Multiply | Multiply the variable x by f(x) to get x * f(x) |
| 4. Integrate | Integrate x * f(x) over the specified range. |
The mean of a probability density function represents the average value of the random variable. It's computed using an integral, as demonstrated in the references, where a numerical result of 1.8484 was obtained by evaluating the integral.
