IID random variables

[Bharti Singla]

Hi all, please anyone help me with this query:
If X and Y are two random variables, then their parameters will be same or not in each of the following cases:
• X and Y are independent and identically distributed (iid).
• X and Y are identically distributed but not independent.

A bit explanation with an example with be helpful!
Thanks in advance.

 

[vgarg]

When two or more random variables are identically distributed then we say each of the random variable has same probability distribution. And same probability distribution means, in case of parametric distributions, that parameters are all same for all the random variables to be identically distributed. They may or may not be independent.
For example,
Let \(W,X,Z \overset{i.i.d}{\sim}\mathcal{N} (\mu, \sigma ^2)\) and \[ Y = \begin{cases} X & \text{ if } W>\mu, \\ Z & \text{Otherwise} \end{cases} \]

Now in order to find the distribution of \(Y\) we can find its distribution function, \[ \begin{align*} F_Y(y) &= \Pr(Y \le y)\\ &= \Pr(Y\le y| W>\mu)\Pr(W>\mu) + \Pr(Y\le y | W\le \mu)\Pr(W \le \mu) \\ &= \Pr(X\le y |W > \mu) \frac{1}{2} + \Pr(Z \le y | W \le \mu)\frac{1}{2} \\ &= \Pr(X\le y) \frac{1}{2} + \Pr(Z \le y )\frac{1}{2} \quad \because X,Z,W \text{ are independent}\\ &= F_X(y)0.5 + F_Z(y)0.5\\ &= F_X(y) \quad \because X,Z \text{ are identically distributed} \end{align*} \]

This means that \(Y \sim \mathcal{N}(\mu, \sigma ^2)\) and \(Y\) is not independent of random variables \(W,X,Z\) but \(X,Y,W,Z\) are identically distributed i.e. same distribution with same parameters.

 

[Bharti Singla]

Very well explained! Thankyou so much.