i.i.d. (independent and identically distributed)

Last updated on 17 Apr 2023

The term "i.i.d." is an acronym for "independent and identically distributed". It is a statistical assumption that describes a set of random variables or observations, and it is widely used in many areas of science and engineering, including finance, economics, and machine learning. In this explanation, we will delve into the meaning of i.i.d. and its importance in statistical analysis.

Random variables and observations

A random variable is a variable whose value is determined by chance. In statistics, we often use random variables to describe the behavior of a population or a sample. For example, suppose we want to study the heights of adult males in the United States. We could define a random variable X to represent the height of a randomly selected male from the population. The value of X could be any real number within a certain range, such as 5 feet to 7 feet.

An observation, on the other hand, is a value that is obtained by measuring or recording a random variable. In our example, an observation of X would be the height of a specific adult male in the United States.

Independence

The first part of the i.i.d. assumption is independence. Two random variables are independent if the value of one variable does not affect the value of the other variable. For example, suppose we flip a fair coin twice. Let X1 be the result of the first flip (either heads or tails), and let X2 be the result of the second flip. The outcomes of the two flips are independent because the result of the first flip has no influence on the result of the second flip.

Independence is an important concept in statistics because it allows us to make assumptions about the relationship between different random variables. If two random variables are independent, we can assume that their joint probability distribution can be factorized into the product of their individual probability distributions. This assumption simplifies the analysis of the data and makes it easier to perform statistical inference.

Identically distributed

The second part of the i.i.d. assumption is identically distributed. Two random variables are identically distributed if they have the same probability distribution. For example, suppose we roll a fair six-sided die twice. Let X1 be the result of the first roll (a number from 1 to 6), and let X2 be the result of the second roll. The two random variables are identically distributed because they both follow the same probability distribution (a uniform distribution over the numbers 1 to 6).

The identically distributed assumption is also important in statistics because it allows us to make assumptions about the population or sample we are studying. If we assume that the random variables in our sample are identically distributed, we can use statistical methods to estimate the parameters of the underlying distribution. For example, we could estimate the mean and variance of the distribution from the sample mean and sample variance.

Independent and identically distributed

When we combine the concepts of independence and identical distribution, we get the i.i.d. assumption. A set of random variables is said to be i.i.d. if each random variable is independent of the others and all have the same probability distribution. In other words, if we have a sample of n observations, X1, X2, ..., Xn, then they are i.i.d. if and only if:

Each Xi is independent of all the other Xj's, where j ≠ i.
Each Xi has the same probability distribution.

The i.i.d. assumption is very useful in statistical analysis because it simplifies the computations and allows us to make strong assumptions about the underlying population. For example, if we have a sample of i.i.d. observations, we can estimate the mean of the population by taking the average of the sample. This is known as the sample mean, and it is an unbiased estimator of the population mean. We can also estimate the variance of the population by taking the sample variance, which is defined as:

s^2 = (1/(n-1)) * ∑(Xi - X̄)^2

where X̄ is the sample mean, n is the sample size, and ∑ represents the sum over all the observations. The sample variance is an unbiased estimator of the population variance, provided that the observations are i.i.d.

Importance of the i.i.d. assumption

The i.i.d. assumption is a fundamental assumption in many statistical models and methods. It allows us to use powerful tools from probability theory and statistics to analyze data and make predictions. Some of the key reasons why the i.i.d. assumption is important are:

Simplifies the analysis: The i.i.d. assumption simplifies the analysis of data by allowing us to treat each observation as an independent and identically distributed sample from the same population. This makes it easier to compute probabilities and statistical measures such as means and variances.
Allows for statistical inference: The i.i.d. assumption allows us to make statistical inferences about the underlying population based on a sample of observations. We can estimate population parameters such as means and variances, and test hypotheses about the population using statistical tests such as the t-test and the F-test.
Enables machine learning: The i.i.d. assumption is also important in machine learning, where it is used to train and evaluate models. Machine learning models are typically trained on a dataset of i.i.d. observations, and their performance is evaluated on a separate set of i.i.d. observations. The i.i.d. assumption allows us to make assumptions about the relationship between the training and test datasets, which is important for ensuring that the model generalizes well to new data.

Limitations of the i.i.d. assumption

While the i.i.d. assumption is a powerful and useful tool in statistical analysis, it is important to recognize its limitations. Some of the key limitations of the i.i.d. assumption are:

Violations of independence: If the observations in a sample are not independent, then the i.i.d. assumption is violated. For example, if we collect data on the heights of family members, then the heights of siblings are likely to be correlated, violating the independence assumption. In such cases, alternative statistical methods may be needed to account for the correlation between observations.
Violations of identical distribution: If the observations in a sample are not identically distributed, then the i.i.d. assumption is also violated. For example, if we collect data on the heights of males and females, then the distributions of the two groups are likely to be different, violating the identical distribution assumption. In such cases, we may need to use different statistical methods for each group or combine the groups in a different way.
Small sample sizes: The i.i.d. assumption is less reliable when the sample size is small. In small samples, there may be too little data to accurately estimate population parameters, and the sample may not be representative of the population. In such cases, we may need to collect more data or use alternative statistical methods that are better suited for small samples.

Conclusion

The i.i.d. assumption is a fundamental concept in statistics that describes the relationship between a set of random variables or observations. The assumption of independence and identical distribution simplifies the analysis of data and allows us to make strong assumptions about the underlying population. However, it is important to recognize the limitations of the i.i.d. assumption and use alternative statistical methods when it is violated.