If X is a random variable and ‘a’ is any constant, then prove that

1.	If X is a random variable and ‘a’ is any constant, then prove that E(ax) = aE(X) and var(ax) = a2var(x).
Answer» Proof : (i) E(ax) = Σax(px) By definition E(X) = ΣxP(X) = aΣxP(X) ∴ E(ax) = aE(x) (ii) var(ax) = E[ax – aE(x)]² by definition of var (x) = E [x – E(x)]² = E [ax – aE(x)]² = a²E [x – E(x)]² var(ax)=a² var(x).

If X is a random variable and ‘a’ is any constant, then prove that E(ax) = aE(X) and var(ax) = a2var(x).

Answer»

Proof : (i) E(ax) = Σax(px)

By definition E(X) = ΣxP(X) = aΣxP(X)

∴ E(ax) = aE(x)

(ii) var(ax) = E[ax – aE(x)]²

by definition of var (x) = E [x – E(x)]² = E [ax – aE(x)]² = a²E [x – E(x)]²

var(ax)=a² var(x).

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If X is a random variable and ‘a’ is any constant, then prove that E(ax) = aE(X) and var(ax) = a2var(x).

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