Meaning of Independent Identically distributed random variables

In summary, Independent Identically distributed random variables refer to a sequence of random variables that are independent and have the same probability distribution function (pdf) and cumulative distribution function (cdf). This means that they also have the same mean and variance. However, just because a set of random variables are independent does not necessarily mean they are also identically distributed. In the example given, the random variables are all normal but have different parameters, so they are not IID, only independent.
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I am a little fuzzy on the meaning of Independent Identically distributed random variables. I understand the independent part but still not 100% on the identically distributed part. I understand that identically distributed means they have the same pdf and cdf but does this mean that they have the same mean and variance?

For example if i have a sequence of random variables: N(0,1),N(2,4),N(3,5) and they are all independent are they IID?
 
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If they have the same pdf, then they by definition have the same mean and variance.
 
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For example if i have a sequence of random variables: N(0,1),N(2,4),N(3,5) and they are all independent are they IID?
They are all normal, but since the parameters are different, the distribution functions are different. (Not IID, only I).
 
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Independent identically distributed (IID) random variables are a fundamental concept in statistics and probability theory. Simply put, IID random variables are a collection of random variables that are independent of each other and have the same probability distribution. This means that each individual random variable in the collection follows the same probability distribution and is not affected by the values of the other variables.

In your example, the three random variables N(0,1), N(2,4), and N(3,5) are independent because the values of one variable do not affect the values of the others. However, they are not IID because they do not have the same probability distribution. The first variable has a mean of 0 and a variance of 1, the second has a mean of 2 and a variance of 4, and the third has a mean of 3 and a variance of 5.

IID random variables are useful because they allow us to make generalizations and predictions based on a sample of data. For example, if we have a sample of IID random variables, we can use the sample mean to estimate the population mean, and the sample variance to estimate the population variance. This concept is essential in many statistical analyses and is the basis for many important statistical tests and models.

In summary, IID random variables are a collection of independent random variables that have the same probability distribution, but they do not necessarily have the same mean and variance. Understanding this concept is crucial for accurately interpreting and analyzing data in many scientific fields.
 

1. What does it mean for random variables to be independent and identically distributed?

Independent and identically distributed (i.i.d.) random variables are a set of random variables that are independent of each other and have the same probability distribution. This means that the outcome of one random variable does not affect the outcome of another, and each variable has the same likelihood of occurring.

2. Why is it important for random variables to be independent and identically distributed?

Having i.i.d. random variables is important because it simplifies statistical analysis. It allows for the use of strong mathematical tools, such as the central limit theorem, making it easier to make predictions and draw conclusions from the data.

3. How can we determine if a set of random variables are independent and identically distributed?

To determine if a set of random variables are i.i.d., we can perform tests such as the Kolmogorov-Smirnov test or the Chi-square test. These tests compare the observed data to the expected distribution and can help us determine if the variables are independent and have the same distribution.

4. What are some examples of independent and identically distributed random variables?

Some examples of i.i.d. random variables include coin flips, rolling a dice, and drawing cards from a deck. In these scenarios, the outcomes of each event are independent of each other and have the same probability of occurring.

5. How does the concept of independent and identically distributed random variables apply to real-world data?

In real-world data, it is often assumed that the data is i.i.d. in order to make statistical analysis easier. However, this assumption may not always hold true, and it is important to carefully assess the data to determine if the variables are truly independent and identically distributed. Failure to do so can lead to inaccurate conclusions and predictions.

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