Meaning of Independent Identically distributed random variables

Click For Summary
Independent Identically Distributed (IID) random variables are those that are both independent and share the same probability distribution. The term "identically distributed" implies that they have the same probability density function (pdf) and cumulative distribution function (cdf), which also means they possess the same mean and variance. In the example provided, the random variables N(0,1), N(2,4), and N(3,5) are independent but not identically distributed due to differing parameters, thus they are not IID. The key takeaway is that while independence is necessary, identical distribution is crucial for the IID classification. Therefore, for random variables to be considered IID, they must have identical statistical properties.
dionysian
Messages
51
Reaction score
1
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?
 
Physics news on Phys.org


If they have the same pdf, then they by definition 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?
They are all normal, but since the parameters are different, the distribution functions are different. (Not IID, only I).
 
Last edited:


Thanks.
 

Thread 'Hypothesis testing: Defining H0, HA hypotheses so that ( H_A)_A' makes sense'

The standard _A " operator" maps a Null Hypothesis Ho into a decision set { Do not reject:=1 and reject :=0}. In this sense ( HA)_A , makes no sense. Since H0, HA aren't exhaustive, can we find an alternative operator, _A' , so that ( H_A)_A' makes sense? Isn't Pearson Neyman related to this? Hope I'm making sense. Edit: I was motivated by a superficial similarity of the idea with double transposition of matrices M, with ## (M^{T})^{T}=M##, and just wanted to see if it made sense to talk...
View full post »

Similar threads

Undergrad How to show these random variables are independent?
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Randomly Stopped Sums vs the sum of I.I.D. Random Variables
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad IID and Dependent RVs: A Closer Look at their Relationship and Parameters
  • · Replies 6 ·
Replies
6
Views
1K
Undergrad What is the expected (squared) coefficient of variation of a sample?
  • · Replies 6 ·
Replies
6
Views
3K
MHB Distribution and Density functions of maximum of random variables
  • · Replies 6 ·
Replies
6
Views
4K
Graduate Sum of independent random variables and Normalization
  • · Replies 10 ·
Replies
10
Views
3K
Undergrad Random variable vs Random Process
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Linear regression and random variables
  • · Replies 30 ·
2
Replies
30
Views
4K
Undergrad Sampling theory and random sample
  • · Replies 5 ·
Replies
5
Views
2K
High School The CDF of the Sum of Independent Random Variables
  • · Replies 10 ·
Replies
10
Views
6K