I am confused about how to approach sequence of random variables that are not identically distributed. For example, consider a sequence $X_1, X_2, \dots, X_n$ with the pdf:

$$ f(X_n)= \begin{cases} (n-1)/2& \text{if }-1/n < x < 1/n\\ 1/n&\text{if }n < x < n+1 \\ 0 &\text{ otherwise} \end{cases} $$

How should i go about finding the mean of $X_n$?

  • $\begingroup$ This is routine bookwork and should carry the self-study tag. Did you try drawing the pdf? In fact this one is so simple you can do it by inspection: there are two uniform components, one with mean 0 and one with mean $n+\frac{1}{2}$. Since the one with mean 0 contributes 0 for its proportion, and the second one has probability $1/n$, the mean is just the product of the mean for that component and its probability. $\endgroup$ – Glen_b Oct 18 '14 at 22:59
  • $\begingroup$ The guidelines are in the self-study tag wiki here, and also briefly mentioned in the first page of the help, which links here. $\endgroup$ – Glen_b Oct 18 '14 at 23:07

As usual.

$$E(X_n) = \int_{-\infty}^{\infty}f_X(x) x dx = \int_{-1/n}^{1/n}\frac{n-1}2 x dx+\int_{n}^{n+1}\frac{1}n x dx$$

$$=...= 1+\frac 1{2n}$$

  • $\begingroup$ Oh.. Straight forward. Another question, when we say that a sequence of random varibles CONVERGES, what exactly do we mean? I mean mathematically the definitions are okay. But what is the meaning? Should this be a separate question? $\endgroup$ – statBeginner Oct 18 '14 at 22:27
  • $\begingroup$ @statBeginner The convergence of random variables is much more nuanced than the convergence of real numbers since random variables are stochastic. This leads to several different definitions of convergence when it comes to random variables, among the most important of which are convergence in distribution, convergence in probability, and convergence almost surely. When you say that a sequence of random variables converges, you must say in what manner (e.g. in distribution, ...). You should see this: en.wikipedia.org/wiki/Convergence_of_random_variables $\endgroup$ – Blue Marker Oct 18 '14 at 22:49
  • $\begingroup$ Asking about intuition related to convergence of random variables -yes, it should be a different question, where it would be good if you included in the question what are your thoughts about it. $\endgroup$ – Alecos Papadopoulos Oct 18 '14 at 23:06

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