I would like to ask you a question - I bumped into a problem that I do not know how to solve- Let $X_1;\dots;X_n$ and $Y_1;\dots; Y_m$, be two random samples from distributions with means $\mu_1$ and $\mu_2$, respectively, and with the same variance $\sigma^2$.

I would like to know how can I calculate $E(X-Y)$ and $\operatorname{ Var} (X-Y)$ - could you give me a hint?

I think that the problem is having insufficient information, would be grateful for any tips. Thanks

  • $\begingroup$ What is the relationship between "$X$" and the $X_i$ and between "$Y$" and the $Y_j$? $\endgroup$ – whuber Nov 9 '15 at 0:35

There are two options:

  1. either the two variables are correlated (or might be), and in this case the sample values must have been collected in pairs, which means that the size of both samples must be the same. The fact that the size of the samples is indexed with two different variables n and m indicates otherwise. Please read this post discussing the calculation of the covariance of two samples with different sizes for further clarification.
  2. or the variables are not correlated, and in this case:

$$E(X-Y)= E(X) - E(Y) = \mu_1 - \mu_2 $$ Regarding the Variance: $$Var(X−Y) = Var(X) + Var(Y) - 2* COV(X,Y) = 2\sigma^2 - 2COV(X, Y)= 2\sigma^2$$ Note that COV(X, Y) = 0 because the two variables are assumed independent

P.S: I am assuming $\mu_1, \mu_2, \sigma^2$ are known or that you know how to estimate them. Let me know if otherwise and I will clarify further

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  • $\begingroup$ Two correlated variables don't have to be sampled in pairs. Regardless, the formula for the expectation of the difference has nothing to do with whether the variables are correlated. $\endgroup$ – whuber Nov 9 '15 at 0:35
  • $\begingroup$ Thanks @whuber. What I meant is that calculating the variance of two correlated variables can only be done if they are sampled in pairs. Would you agree with that statement? $\endgroup$ – Ihab Nov 9 '15 at 0:45
  • $\begingroup$ I cannot agree or disagree yet because I still don't know what the question is asking. It is phrased in terms of random variables and unspecified "samples," so conceivably it is asking for some kind of mathematical calculation. That wouldn't necessarily require a pairing. $\endgroup$ – whuber Nov 9 '15 at 0:48
  • $\begingroup$ @whuber If there are 10 observations of $X$ and 30 observations of $Y$, are you suggesting that these observations can be used to estimate $cov(X,Y)$? $\endgroup$ – Dave Sep 19 '19 at 2:09
  • $\begingroup$ @Dave My concern four years ago is reflected in the comment I posted beneath the question. In response to your question, it occurs to me that there are various ways one could estimate a covariance without having paired data. For instance, observing a third variable $Z$ simultaneously with $X$ and simultaneously with $Y$ could provide useful information, depending on what model one adopts. There is also the possibility of observing linear combinations of $X$ and $Y.$ For instance, from a sample of $X-Y$ and an independent sample of $X+Y$ one may easily estimate $\operatorname{Cov}(X,Y).$ $\endgroup$ – whuber Sep 19 '19 at 13:21

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