Given 𝑛 independent Normally distributed random variables 𝑋ᵢ ∼ 𝑁(𝜇ᵢ,𝜎²ᵢ) and 𝑛 real constants 𝑎ᵢ∈ℝ, I need to find an acceptable Normal approximation of the distribution of 𝑌 random variable (assuming Pr[𝑋ᵢ≤0]≈0, to avoid divisions by zero)
Y = ∑aᵢXᵢ / ∑Xᵢ
I thought to split 𝑌 into single components
Y = a₁X₁ / ∑Xᵢ + a₂X₂ / ∑Xᵢ + ... + aₙXₙ / ∑Xᵢ
Y = a₁Y₁ + a₂Y₂ + ... + aₙYₙ
where the distribution of each 𝑌ᵢ can be found noting that
Yᵢ = Xᵢ / (Xᵢ + ∑Xⱼ) , for j≠i
and that
1/Yᵢ = (Xᵢ + ∑Xⱼ) / Xᵢ = 1 + ∑Xⱼ / Xᵢ
so, calling ∑Xⱼ = Uᵢ we can say that Xᵢ and Uᵢ are independent and, according to Díaz-Francés et Al. 2012, a Normal approximation of 1/𝑌ᵢ can be the one in figure 1 and, considering 1 ~ N(1,0), the r.v. Yᵢ can be approximated to figure 2. Thus, approximation of each aᵢYᵢ is the one in figure 3.
But now... I'm stuck at the sum of aᵢYᵢ because, not being independent, I don't know how to approximate the variance of their sum.
Any advice? Any more straightforward or more efficient method?
The sum of Normal variables is a Normal variable with mean the sum of the means and variance the sum of variances. So Y is the ratio of two Normal variables. Y = N1/N2. The ratio of two Normal variables is distributed as a Cauchy distribution:
- https://en.wikipedia.org/wiki/Ratio_distribution
- https://mathworld.wolfram.com/NormalRatioDistribution.html
Hi Luca. Thanks for your answer.
These results are for independent r.v. only, aren't they?
In my case, numerator is not independent from denominator.
I will hazard a guess: we know that the sum of many independent random variables will be normally distributed (no matter their distribution). We know that the sum of many 100% dependent variables will have the same distribution as any one of them. So the sum of many variables with some degree of correlation will tend towards normally distributed, but the rate at which it converges to the normal distribution with increasing number of variables is dependent on the degree of correlation between the variables. You should find that the sum of Normally distributed variables that are correlated with each other is still distributed normally.
Hi Timothy A Ebert
Thanks for your answer.
Yes, the sum of all aᵢYᵢ will be "roughly" normal or, at least, will tend to a normal distribution as n increases. The challenging part is to find the correct variance approximation of the sum.
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