Finding the mean using the CDF

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In summary, the conversation discusses finding the mean of a function in terms of a complicated random variable. By using integration by parts, the mean expression can be simplified to \int_0^{\infty}e^{-x}F_X(x)\,dx, which is mathematically correct.
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EngWiPy
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Hello all,

I have the random variable ##X## with CDF and PDF of ##F_X(x)## and ##f_X(x)##, respectively. Now I have a function in terms of the random variable ##X##, which is ##e^{-X}##, and I want to find the mean of this function. Basically this can be found as

[tex]\int_0^{\infty}e^{-x}f_X(x)\,dx[/tex]

where the range of the random variable ##X## is between ##0## and ##\infty##. However, ##X## is a very complicated random variable (it's a function of a number of other independent random variables), and thus although the CDF is easy to find relatively, the PDF is not that nice expression after taking the derivative of the CDF function. So someone here pointed out to me that by using integration by parts, the above mean expression can by written as

[tex]\int_0^{\infty}e^{-x}f_X(x)\,dx=\left. -e^{-x}F_X(x)\right|_0^{\infty}+\int_0^{\infty}e^{-x}F_X(x)\,dx=\int_0^{\infty}e^{-x}F_X(x)\,dx[/tex]

I just wanted to make sure: Is this mathematically correct?

Thanks
 
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Anyone on this?
 
  • #3
Looks good to me.
 
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What is the CDF method for finding the mean?

The Cumulative Distribution Function (CDF) method is a statistical tool used to calculate the probability of a random variable falling within a certain range of values. In the context of finding the mean, the CDF method involves calculating the area under the curve of a probability distribution function up to a specific value, and then dividing by the total area under the curve.

Why is the CDF method used to find the mean?

The CDF method is often used to find the mean because it takes into account the entire distribution of data and is not affected by outliers or extreme values. This method provides a more accurate representation of the central tendency of a dataset compared to other methods, such as using the arithmetic mean.

What types of data can be analyzed using the CDF method?

The CDF method can be used to analyze any type of data that follows a continuous distribution, such as normal, exponential, or uniform distributions. It can also be applied to discrete distributions by using a cumulative sum instead of a cumulative integral.

How is the mean calculated using the CDF method?

To calculate the mean using the CDF method, the area under the curve of the probability distribution function is divided by the total area under the curve. This can be represented mathematically as:

Mean = ∫ab x*f(x) dx / ∫ab f(x) dx

What are the advantages of using the CDF method over other methods for finding the mean?

One of the main advantages of using the CDF method is that it provides a more robust measure of central tendency compared to other methods. It also takes into account the entire distribution of data, making it less affected by outliers. Additionally, the CDF method can be used for a wide range of data types and distributions, making it a versatile tool for statistical analysis.

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