Finding expectation of peicewise mixed distribution density function

In summary, the expected value of a piecewise mixed distribution density with CDF starting from a and ending at b can be found by differentiating each piece of the CDF and using the definition of expectation. If the function is in a non-integrable form, the deltas can be used for discrete distributions or the properties of the characteristic function can be used for Lebesgue forms.
  • #1
torquerotates
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If I am given the CDF of a piecewise mixed distribution density starting from a and ending at b, would the expected value just be a + integral(all the pieces) ?
 
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  • #2
If there were only one "piece" would the expected value be a+ the integral of that piece?
 
  • #3
I suppose not. It would just be the integral of that piece. But if it were two or more?
 
  • #4
torquerotates said:
If I am given the CDF of a piecewise mixed distribution density starting from a and ending at b, would the expected value just be a + integral(all the pieces) ?

Hey torquerotates.

Why don't you just differentiate each piece of your CDF and then use the definition of expectation?

If you function is in discrete form or a form that is non-integrable (non-Riemannian) then you can either just look at the deltas if you have a discrete distribution for that piece, or if its in a Lebesgue form just use the properties of the characteristic function to get your PDf for that piece.
 
  • #5


Yes, the expected value of a piecewise mixed distribution density function can be calculated by taking the integral of the function over the entire range from a to b. This is because the expected value is a measure of the central tendency of the distribution, and the integral represents the average value of the function over its entire range. However, it is important to note that the integral must also take into account any weights or probabilities associated with each piece of the distribution, as these can affect the overall expected value. Additionally, the integral may need to be adjusted if the function is not continuous at certain points within the range.
 

1. What is a piecewise mixed distribution density function?

A piecewise mixed distribution density function is a mathematical function used to describe the probability distribution of a random variable. It consists of multiple sub-functions, each defined over a different interval, and the sub-functions are pieced together to form the overall function. This type of function is commonly used when the underlying data has different characteristics in different regions.

2. How do you find the expectation of a piecewise mixed distribution density function?

The expectation of a piecewise mixed distribution density function can be found by taking the weighted average of the expectations of each sub-function. This involves multiplying the expectation of each sub-function by its corresponding weight (the proportion of the data in that interval), and then summing these values together.

3. What is the importance of finding the expectation of a piecewise mixed distribution density function?

Calculating the expectation of a piecewise mixed distribution density function is important because it gives us a measure of the central tendency of the data. It allows us to make predictions about the behavior of the random variable and can be used to compare different distributions.

4. What are some common applications of piecewise mixed distribution density functions?

Piecewise mixed distribution density functions are commonly used in statistics and probability to model a wide range of phenomena. Some examples include modeling the distribution of income, the distribution of weather patterns, and the distribution of stock returns.

5. Are there any limitations to using piecewise mixed distribution density functions?

One limitation of using piecewise mixed distribution density functions is that they can be difficult to interpret and calculate, especially when there are many sub-functions involved. Additionally, they may not accurately represent the underlying data in some cases, and there may be other types of distribution functions that are better suited for certain situations.

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