How Do You Prove Independence of Z-Squared in Sample Distributions?

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In summary, the question is asking for help with understanding how to find the distribution of the sum of two independent chi square distributions with different degrees of freedom. The individual samples are assumed to be independent. The solution involves utilizing the moment generating function and applying the property that the sum of two chi square distributions with degrees of freedom k1 and k2 is another chi square distribution with degrees of freedom k1 + k2.
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fighthard88
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Homework Statement



[PLAIN]http://img40.imageshack.us/img40/1503/question1l.jpg

Homework Equations



[URL]http://onlinecourses.science.psu.edu/stat414/sites/onlinecourses.science.psu.edu.stat414/files/lesson26/Variance10.gif[/URL]

[URL]http://onlinecourses.science.psu.edu/stat414/sites/onlinecourses.science.psu.edu.stat414/files/lesson26/Variance11.gif[/URL]

[URL]http://onlinecourses.science.psu.edu/stat414/sites/onlinecourses.science.psu.edu.stat414/files/lesson26/Variance13.gif[/URL]

The Attempt at a Solution



I need help with this question. I know that to get this distribution, I need to sum the Z^2's of both respective samples. However, in order to do so wouldn't I need to prove that Z^2's are independent? I'm assuming I'll need to utilize the moment generating function. However, I'm not sure how to go about this. Any help would be much appreciated!
 
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so if you're happy up to the point where Z^2 is equivalent to chi square distribution with 1 DoF, then the sum of two chi square distribution with DoF k1 & k2 is another chi square distribution with DoF = k1 + k2.

the question says the 2 samples are independent. As it is not stated otherwise I would assume the individual samples are independent, though state it as a n assumption
 

FAQ: How Do You Prove Independence of Z-Squared in Sample Distributions?

What is a sample distribution question?

A sample distribution question is a type of statistical question that involves analyzing a set of data from a sample (a smaller group) in order to make conclusions about a larger population.

How is a sample distribution different from a population distribution?

A sample distribution represents a smaller group of data that has been collected from a larger population. The population distribution represents the entire group of data.

Why is it important to understand sample distribution?

Understanding sample distribution allows researchers to make inferences about a larger population without having to collect data from every individual in that population. It also helps to identify patterns and trends within the data.

What are some common methods for analyzing sample distribution?

Some common methods for analyzing sample distribution include calculating descriptive statistics such as mean, median, and standard deviation, as well as conducting inferential statistics tests such as t-tests and ANOVA.

How can sample distribution affect the accuracy of research findings?

Sample distribution can significantly impact the accuracy of research findings. A sample that is not representative of the larger population or is too small may lead to biased or inaccurate conclusions. It is important to ensure that the sample is properly selected and accurately represents the population in order to obtain reliable results.

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