Query regarding Independent and Identically Distributed random variables

In summary, the conversation discusses a question about i.i.d. random variables and how to prove that the variance of a sequence of these variables, divided by n, is equal to the variance of each individual variable. The experts suggest two methods to prove this, both relying on the independence and common variance of the variables.
  • #1
maverick280857
1,789
4
Hi

I have a question regarding i.i.d. random variables. Suppose [itex]X_1,X_2,\ldots[/itex] is sequence of independent and identically distributed random variables with probability density function [itex]f_{X}(x)[/itex], mean = [itex]\mu[/itex] and variance = [itex]\sigma^2 < \infty[/itex].

Define

[tex]Y_{n} = \frac{1}{n}\sum_{i=1}^{n}X_{i}[/tex]

Without knowing the form of [itex]f_{X}[/itex], how does one prove that [itex]var(Y_{n}) = \sigma^2/n[/itex]?

I suppose this is a standard theorem/result, but any hints/ideas to prove this would be appreciated.

Thanks.
 
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  • #2
var(Yn)=E(Yn2)-E(Yn)2

Plug in the series for Yn and expand, using the fact the E(sum)=sum(E) and E(prod of ind. rv)=prod of E's., it will all work out. Note that all you needed was independence and the fact that the mean and variance was the same for all. The distributions could have been different.
 
  • #3
Thanks mathman :smile:
 
  • #4
Can also be done as follows:

If [itex]T = \sum_{i=1}^{n}a_{i}X_{i}[/itex] then [itex]Var(T) = \sum_{i=1}^{n}a_{i}^2Var(x_{i})[/itex], which gives

[tex]Var(Y_{n}) = \sum_{i=1}^{n}\frac{1}{n^2}Var(x_{i}) = \frac{\sigma^2}{n}[/tex]

Edit: need only the independence of the random variables
 
Last edited:

Related to Query regarding Independent and Identically Distributed random variables

1. What is the concept of independent and identically distributed random variables?

Independent and identically distributed random variables refer to a set of random variables that are independent of each other and have the same probability distribution. This means that the outcome of one variable does not affect the outcome of another, and they all follow the same probability distribution.

2. How are independent and identically distributed random variables used in statistics?

Independent and identically distributed random variables are commonly used in statistical analyses to model and analyze a wide range of data. They are often used in hypothesis testing, regression analysis, and in constructing confidence intervals.

3. What is the significance of independence in independent and identically distributed random variables?

Independence is a crucial aspect of independent and identically distributed random variables as it ensures that the variables are not influenced by each other, allowing for accurate analysis and interpretation of the data. It also allows for the application of various statistical methods and techniques.

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

Some real-life examples of independent and identically distributed random variables include flipping a coin multiple times, rolling a dice, and drawing cards from a deck. In all of these situations, each outcome is independent of the previous one and follows the same probability distribution.

5. What is the difference between independent and identically distributed random variables and independent and non-identically distributed random variables?

The main difference between these two types of random variables is that in independent and identically distributed random variables, the variables are independent of each other and follow the same probability distribution, whereas in independent and non-identically distributed random variables, the variables are independent but may follow different probability distributions.

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