# On the requirement of the word "identical" in the phrase "Independent but not Identically distributed"?

Assuming that I have 3 independent exponential random variable $$X_1,X_2,X_3$$ with parameters $${\lambda _1},{\lambda _2},{\lambda _3}$$, it is trivial that when $${\lambda _1} = {\lambda _2} = {\lambda _3}$$ I can immediately call them Independent Identically Distributed.

Question 1, in the case $${\lambda _1} \ne {\lambda _2} = {\lambda _3}$$, can I say that $$X_1,X_2,X_3$$ are independent but not identically distributed ?

Question 2, again if I have three independent random variable $$X_1,X_2,X_3$$ that is not exponential anymore but for example Binomial, Gaussian, Gamma, Can I said they are independent non-identically distributed ?

Question 3, does independent but not identically distributed require the independent random variable to have the same distribution but difference in parameters ?

Edit: added the word independent to avoid confusion

• It is not true that all exponential random variables $X_1,X_2,X_3$ with $\lambda_1=\lambda_2=\lambda_3$ will be independent. They will in that scenario be identically distributed but independence requires more information specifically about how the random variables interact with one another... not how they act by themselves. Commented Mar 4, 2021 at 12:56
• Q1, assuming they are independent, yes they are independent but not identically distributed... but that "assuming they are independent" is a big if. Q2, yes again with the same disclaimer. Q3 No, independence has nothing to do with the distribution types... you can have one be of one distribution type and the other be of a different type and yet still be independent. Independence of random variables has nothing to do with distribution types, it exclusively has to do with whether or not $\Pr(X_1=x_1,X_2=x_2)=\Pr(X_1=x_1)\cdot\Pr(X_2=x_2)$ for all $x_1,x_2$ or not which is in general untrue Commented Mar 4, 2021 at 12:58

The phrase "independent but not identically distributed" simply means that the random variables are independent, but they do not follow the same distribution. Here "the same distribution" refers to the same type of distribution with the same parameters. For example, if $$X_1,X_2,X_3$$ are independent and $$X_1,X_2\sim B(5,0.5),\:X_3\sim B(5,0.6)$$ then they are independent but not identically distributed. If $$Y_1,Y_2,Y_3$$ are independent and $$Y_1,Y_2\sim \text{Po}(5),\:Y_3\sim N(0,1)$$ then they are also independent but not identically distributed.