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Let $x_i$ be $n$ i.i.d random variables from a uniform distribution in the interval $[a,b]$. Let $y_i$ be i.i.d random variables also from a uniform distribution in the interval $[c,d]$. Also, the pairs $(x_1,y_1),\dots,(x_n,y_n)$ are independent. Individual $x_i$ and $y_i$ may or may not be correlated. Consider the random variables obtained as \begin{align} z_1 \,&=\,\frac{1}{n}\sum_{i=1}^{n}\frac{x_i}{y_i} \\ z_2 \,&=\,\frac{\sum_{i}^{n}x_i}{\sum_{i}^{n}y_i} \end{align}

  • Is there any interesting aspects/relationships between the means of these random variables?
  • This comes from a practical problem and it is very reasonable to assume that $n$ is very large compared to $a,b,c,d$. Thus, we can consider questions in the limit when $n$ tends to infinity. Are there any interesting insights there?
  • Consider the cases where $x_i$ and $y_i$ have strong positive correlation for every $i$. Is there any additional insights in that scenario to the above questions?
dineshdileep
  • 8,887
  • For $z_2$, expand the fraction with $1/n$ in numerator and denominator. Just to be clear: do you assume that all $x_i,y_i$ are jointly independent? Your last point seems to indicate that you don't want to assume this. Do you know that the $(x_1,y_1),(x_2,y_2),\dots$ are independent? – PhoemueX Aug 01 '22 at 20:20
  • @PhoemueX thanks for pointing that out. Let me edit the question. – dineshdileep Aug 02 '22 at 01:06
  • You likely want $0<c<d$. If so, and if ${(X_i, Y_i)}_{i=1}^{\infty}$ are i.i.d., then $z_1(n)\rightarrow E[X_1/Y_1]$ almost surely and $z_2(n)\rightarrow E[X_1]/E[Y_1]$ almost surely (by the Law of Large Numbers). – Michael Aug 02 '22 at 07:17
  • Here is a related question: https://math.stackexchange.com/questions/2762574/ratio-of-averages-vs-average-of-ratios/2763664#2763664 – Michael Aug 02 '22 at 16:20

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