Concept guide
Variance in finite probability models: translation, scaling and the zero-variance characterization
In a valid finite probability model, prove Var(X)≥0, Var(X+c)=Var(X), Var(kX)=k²Var(X), and Var(X)=0 ⇔ X takes one common value at every positive-weight outcome. Then with S={1,2,3,4}, w₁=1/2, w₂=1/4, w₃=1/4, w₄=0 and X=(1,3,−1,10), compute μ=1, Var(X)=2, Var(X+5)=2 and Var(−3X)=18, and use Z=(2,2,2,7) to show that a different value at a zero-weight outcome does not prevent zero variance.