Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

If two random variables $x$ and $y$ are identical and independently distributed, do they need to have the same mean and variance? Can there exist a case where they are iid and still have different parameters?

share|improve this question
add comment

4 Answers

up vote 1 down vote accepted

If two variables are iid , then they must have the same distribution.Which means they have the same parameters namely mean, std dev.

share|improve this answer
add comment

The mean and variance are determined by the distribution. Thus, if they have the same distribution, they must have the same mean and variance.

share|improve this answer
add comment

If $X$ and $Y$ are iid random variables with cumulative distribution function $F(t)=\mathbb{P}(X\leq t)=\mathbb{P}(Y\leq t)$ and if mean and variance of the variables exists, then $$\mbox{E}X=\int_\mathbb{R}tdF(t)$$ and $$\mbox{Var}X=\mbox{E}X^2-(\mbox{EX})^2=\int_\mathbb{R}t^2dF(t)-(\mbox{E}X)^2.$$ If $F(t)$ is differentiable then you can write $\int_\mathbb{R}tdF(t)$ as $\int_\mathbb{R}tf(t)dt$ where $f(t)$ is a density function of $X$ and $Y$. So if idd variables are defined with such distribution functions, then there is no difference in means and variances. But it might happen that your variable does not have variance or mean. One example is Cauchy random variable, then you can't say anything about the mean or variance, because those characteristics simply do not exist.

share|improve this answer
add comment

The abbreviation iid usually refers to independent and identically distributed" (in linear modelling, error distributions need to be iid)

  • NOT identical and independently distributed (as Daniel's question mentions).

Two variables are "independent" if they are not correlated

Two variables are identically-distributed if they have the same frequency distribution (the same skew/kurtosis/variance etc.)

Two variables are identical if they are exactly the same. This would mean that these variables would be perfectly correlated. Therefore, by definition, they cannot be "independently distributed".

In other words, " identical and independently distributed" is a contradiction in terms.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.