Difference among the same distribution , identical distribution and similar distribution.

Question

$X\sim N(\mu_1,\sigma)$ and $Y\sim N(\mu_2,\sigma)$ are similar but not identical.

$X\sim N(\mu,\sigma)$ and $Y\sim N(\mu,\sigma)$ are identical.

But what is same distribution?

Do same and identical exactly have the same meaning?

Answer 1

Similar distribution means the type of distribution is the same.

Identical distribution means the type of distribution is the same and their parameters have exactly the same value.

If question stated that X and Y have same distribution then their parameters should have same values.

But if question stated that X and Y have same type of distribution that's means their parameters may not have same values.

Answer 2

Same and identical mean the same word in the english language, so they should mean the same in maths too. however, "similar" and "identical" do not mean the same in the english language, so naturally in math they don't mean the same either.

The word "similar" suggests there are resemblances but the two similar things are not all the same ; here by similar we mean that the type of distribution (e.g. normal in your case) is the same, but the parameters may differ. The word "identical" suggests what it suggests, i.e. $X = Y$ as random variables, e.g. in your example $\mathbb P(a \le X \le b) = \mathbb P(a \le Y \le b)$ for all $a,b \in \mathbb R$.

The words "as random variables" is important, because for two dice $A$ and $B$, you can take the variable $X$ which is the result of a throw of dice $A$ and call $Y$ the result of the throw of dice $B$. If your dice are normal six-sided non-tricked dice, then the probability that $X$ takes an integer value $a$ between $1$ and $6$ should be $1/6$, and similarly the probability that $Y$ takes an integer value $b$ between $1$ and $6$ is also $1/6$. So the random variables are equal, but that does not mean that when I throw dice $A$ and $B$ I will always get the same result on both dice. It just means that the probabilities are the same in both cases.

Hope that helps,