# Are these variables mutually independent?

I have a random variable denoted $N$. Then I have random variables denoted $X_1$, $X_2$, ..., $X_N$ distributed according to a uniform distribution. I have also random variables denoted $Y_1$, $Y_2$, ..., $Y_N$ distributed according to this same probability distribution.

If we consider the variable $S_X$ such that $S_X=\sum_{i=1}^{N}X_i$ and $S_Y$ such that $S_Y=\sum_{i=1}^{N}Y_i$.

Are $S_X$ and $S_Y$ mutually independent ?

If we extend this problem to the case of any number $A$ of sums, for instance $S_X$, $S_Y$ and $S_Z$ when $A=3$. Are these $A$ variables mutually independent ?

Thank you very much.

• You might like to take a look at the law of total covariance, which in your case is: $\text{cov}(S_X,S_Y)=\text{E}(\text{cov}(S_X,S_Y \mid N))+\text{cov}(\text{E}(S_X\mid N),\text{E}(S_Y\mid N))$. Pay particular attention to the second term, which is a covariance of two functions of $N$. Jul 11, 2013 at 5:51

Even assuming the $$X_i$$ and $$Y_j$$ are independent of $$N$$ and of one another, $$S_X$$ and $$S_Y$$ will be positively correlated and (therefore) not independent. We can calculate that correlation given the distribution of $$N$$.

To illustrate, let $$N$$ take on either the value $$1$$ or $$12$$ with equal probability. Then half the time $$(S_X, S_Y)$$ is a point in the unit square (when $$N=1$$) and half the time it is a point near $$(6,6)$$ and--according to the Central Limit Theorem--is dispersed randomly around that location in an approximately bivariate Normal manner.

This scatterplot depicts a simulation of 10,000 independent $$(S_X, S_Y)$$ pairs. The colors distinguish the values of $$N$$.

The correlation should be obvious in this plot: $$S_X$$ and $$S_Y$$ both tend to be small when $$N$$ is small and large when $$N$$ is large, whence they tend to be small together or large together: that's positive correlation.

The correlation can be computed using formulas for nested (iterated) expectations. For example,

$$\mathbb{E}_{N;X_i}[S_X] = \mathbb{E}_N[\mathbb{E}_{X_i|N}[S_X | N]] = (1/2 + 12(1/2))/2 = 13/4.$$

In a similar manner all relevant multivariate moments of $$(S_X, S_Y)$$ can be computed based on knowing that $$\mathbb{E}[X_i] = \mathbb{E}[Y_j] = 1/2$$ and $$\mathbb{E}[X_i^2] = \mathbb{E}[Y_j^2] = 1/3$$ (if we assume the $$X_i$$ and $$Y_j$$ are all independent). The variances are $$56/3 - (13/4)^2 \approx 8.104$$ and the covariance is $$145/8 - (13/4)^2 = 7.5625$$, whence the correlation is $$363/389 \approx 0.9332$$. Indeed, in this simulation the observed correlation was $$0.9316$$, apparently differing from this theoretical value only by chance variation.

This answer obviously extends to more than two such sums. It provides a nice example of variables that can be conditionally independent (which will be the case when the $$X_i$$ are independent of the $$Y_j$$) but not themselves independent.

### Simulation Code

The simulation was carried out in R:

N <- 10^4
m <- 12
n <- ifelse(runif(N) < 1/2, 1, m)
x <- matrix(runif(m*N), ncol=m)
y <- matrix(runif(m*N), ncol=m)

s <- t(sapply(1:N, function(i) c(sum(x[i, 1:n[i]]), sum(y[i, 1:n[i]]))))
col = ifelse(n==1, "Blue", "Red")
plot(s, col=col, pch=19, cex=.5, xlab="X", ylab="Y")
cor(s)

• Thank you very much for this simulation. It's more than I could expect. Jul 10, 2013 at 18:24