Identically distributed random variables

Identically distributed random variables have the same probability distribution in Intro to Probability. That means they share the same mean, variance, and overall shape, even if their outcomes are different.

What are Identically distributed random variables?

In Intro to Probability, identically distributed random variables are variables that follow the same distribution. If you know the probability rule for one of them, you know the rule for all the others, even though the actual observed values can differ from trial to trial.

That phrase does not mean the variables are equal or tied to the same outcome. It means they come from the same probabilistic setup. For example, if $X_1, X_2, \dots$ represent repeated measurements from the same process, and each one has the same distribution, then they are identically distributed. Their means match, their variances match, and the full shape of the distribution matches too.

This idea shows up a lot with repeated sampling. A class of variables might all measure the same kind of event, like the result of repeated coin flips, the number of calls in repeated time blocks, or repeated draws from a population under the same conditions. The variables can be different random outcomes, but the long-run behavior is governed by the same distribution each time.

A common mistake is to confuse identically distributed with independent. Identically distributed only says the distributions match. Independence says one variable does not affect the others. You can have variables that are identically distributed but dependent, and you can also have independent variables with different distributions.

The connection to variance is one reason this term matters in the course. If random variables are identically distributed, then each one has the same variance, which makes sums and averages easier to analyze. When the variables are also independent, you get the cleaner variance rules used in limit results like the weak and strong laws of large numbers.

Why Identically distributed random variables matter in Intro to Probability

This term is the setup for a lot of the big theorems in Intro to Probability, especially the laws of large numbers. Those results usually talk about averages of repeated random variables, and the phrase identically distributed tells you that each trial is coming from the same probability model.

That matters because the average only behaves predictably when the pieces being averaged are comparable. If one random variable comes from a coin-flip model and another comes from a different process with a different mean or spread, the average does not have the same clean interpretation. Identical distribution keeps the model homogeneous.

It also makes variance calculations much simpler. If $X_1, \dots, X_n$ are independent and identically distributed, then each variable has the same variance, so the variance of the sum is just $n$ times that common variance. That is one of the standard moves in probability homework: identify the distributional match, then simplify the algebra.

In simulation work, identically distributed random variables are what let you model repeated trials under the same conditions. In proofs and theoretical questions, the term signals that you can use shared mean and variance properties instead of treating each random variable separately.

How Identically distributed random variables connect across the course

Independent random variables

Independence and identical distribution are separate ideas. Independence says one random variable does not change the probability law of another, while identical distribution says the variables have the same distribution. A lot of theorems in probability need both at once, but you should not assume one from the other.

Variance

If random variables are identically distributed, they all have the same variance. That makes variance calculations cleaner when you add them up, especially if they are independent. The shared variance is one of the first things you use when working with repeated trials or sample averages.

Expectation

Identically distributed random variables also share the same expected value. That is why averages of repeated observations can be compared to a single common mean. In limit theorems, the expected value is the number the sample average tends to settle around.

Convergence in Probability

When you study sample averages of identically distributed random variables, convergence in probability is the language of the weak law of large numbers. It tells you the average gets closer and closer to the common expected value as the number of trials grows.

Are Identically distributed random variables on the Intro to Probability exam?

A probability problem set will usually ask you to spot whether a collection of random variables shares the same distribution before you use a formula. That might mean checking whether repeated trials have the same success probability, the same parameter, or the same distribution family. Once you identify that the variables are identically distributed, you can use the shared mean or variance instead of writing a separate expression for each one.

For limit-theorem questions, you often need to say why a sample average is eligible for the weak law of large numbers. If the variables are independent and identically distributed, you can move from the model description to the conclusion that the average stabilizes around the common expectation. On quizzes, a common trap is a setup where each trial looks similar but one parameter changes, which breaks identical distribution.

Identically distributed random variables vs Independent random variables

These are easy to mix up, but they mean different things. Independent random variables do not influence each other, while identically distributed random variables have the same distribution. You can have one without the other. A pair of variables can be independent but not identically distributed, or identically distributed but dependent.

Key things to remember about Identically distributed random variables

• Identically distributed random variables have the same probability distribution, so they share the same mean, variance, and distribution shape.

• The term does not say the variables are equal or independent, it only says they come from the same distribution.

• This idea shows up most often when you model repeated trials, repeated measurements, or sample averages in Intro to Probability.

• If the variables are also independent, variance and law of large numbers calculations become much easier.

• A good check is whether every trial uses the same distributional rule or whether one of the parameters changes from trial to trial.

Frequently asked questions about Identically distributed random variables

What is identically distributed random variables in Intro to Probability?

They are random variables that all follow the same probability distribution. That means they have the same mean, variance, and shape, even though the actual outcomes can be different. The term is common in repeated-trial models, where each trial uses the same probabilistic rule.

Are identically distributed random variables the same as independent random variables?

No. Identical distribution describes the shape of the distribution, while independence describes whether one variable affects another. You often see both together in the same theorem, but they are separate conditions. A variable can satisfy one and not the other.

How do I know if random variables are identically distributed?

Check whether each variable has the same distribution family and the same parameters. If one variable has a different success probability, mean, or rate parameter, then they are not identically distributed. In homework, the setup usually makes this visible through repeated identical trials.

Why do identically distributed random variables matter for the law of large numbers?

Because the law of large numbers studies averages of repeated random variables that come from the same model. If the variables are identically distributed, the average is comparing like with like, so it can settle around a common expected value. Without identical distribution, the average may not have the same clean limit behavior.