Siemensueqw

2022-11-05

Determine the covariance and correlation for ${X}_{1}$ and ${X}_{2}$ in the joint distribution of the multinomial random variables ${X}_{1},{X}_{2}$ and ${X}_{3}$ in with ${p}_{1}={p}_{2}={p}_{3}=\frac{1}{3}$ and n = 3. What can you conclude about the sign of the correlation between two random variables in a multinomial distribution?

Waldruhylm

Given: Multinomial random variables ${X}_{1},{X}_{2},{X}_{3}$
${p}_{1}={p}_{2}={p}_{3}=\frac{1}{3}$
n=3
The correlation measures the strength of the linear relationship between two variables.
When one variable increases as the other variables increases, then the correlation is positive. When one variable decreases as the other variables increases, then the correlation is negative.
Since we require ${X}_{1}+{X}_{2}+{X}_{3}=3$ for a multinomial distribution, one random variable will decrease when another random variable increases and thus the correlation needs to be negative.
This then implies that the sign of the correlation is a minus sign.
Result:
(negative)

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