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Can we have a summation of $n$ stochastic variables equal to a constant? For example, I have $n$ normal variables which are the time for using different functions of the cell phone of one person. So the total using time is a constant, but the time for using different function is a stochastic variable.

FredH
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Dinh Hanks
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  • Yes you can. Let $X_1,X_2,\dots,X_{n-1}$ be any random variables, and let $X_n=1-(X_1+X_2+\dots+X_{n-1})$. Then $X_1+X_2+\dots+X_n=1$ is constant. The situation you describe is a degenerate multivariable normal distribution, to give you something to Google. – Mike Earnest Mar 05 '19 at 22:28
  • Thank you very much for your answer. – Dinh Hanks Mar 06 '19 at 09:21
  • Can i have one more question. Suppose X1,X2,…,Xn−1 is normal distribution, in that way, Xn=1−(X1+X2+⋯+Xn−1) is also normal distribution. Suppose Y1, Y2, ..., Yn are normal distribution. Does summation of Xi.Yi, i=1, 2, ..., n is normal distribution? Thank you very much. – Dinh Hanks Mar 11 '19 at 10:35
  • You should ask that as a separate question. – Mike Earnest Mar 11 '19 at 20:48

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A sum of independent random variables cannot be a constant unless each of them is a constant. Without independence the sum can be a constant: take $X$ and $-X$.

Kavi Rama Murthy
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