Construct finite independent sums and compare repeated copies with scaling
A finite random variable X has the following distribution. The variables X₁,...,Xₙ are mutually independent and each has exactly this distribution
| \(x\) | \(\Pr(X=x)\) |
|---|---|
| \(-3\) | \(\frac{2}{9}\) |
| \(-1\) | \(\frac{4}{9}\) |
| \(1\) | \(\frac{1}{3}\) |
\[n=5,\qquad S_n=X_1+\cdots+X_n\]
Also define \(T=5X\) using just one draw of X, multiplied by 5. T reuses that same value; it is not a sum of independent draws