Question

Let \(c\) be a constant. Show that (a) \(\operatorname{Var}(c X)=c^{2} \operatorname{Var}(X)\) (b) \(\operatorname{Var}(c+X)=\operatorname{Var}(X)\).

Short answer

We proved the properties as follows: (a) \(Var(cX) = c^2 Var(X)\): 1. Use the definition of variance with cX. 2. Apply the linearity of expectation. 3. Factor out the constant from the expectations. 4. Factor out the constant from the entire expression. (b) \(Var(c+X) = Var(X)\): 1. Use the definition of variance with c+X. 2. Expand the squared term and apply the linearity of expectation. 3. Apply the linearity of expectation to each term. 4. Simplify the expression by cancelling out equivalent terms.

Step-by-step solution

Use the definition of variance with cX

We will replace X with cX in the definition of variance to find the variance of cX. \(Var(cX) = E[(cX)^2] - (E[cX])^2\)

Apply the linearity of expectation

We will apply the linearity of expectation to both \(E[(cX)^2]\) and \(E[cX]\). \(Var(cX) = E[c^2 X^2] - (cE[X])^2\)

Factor out the constant from the expectations

We can factor the constant c out of the expectations, remembering that c is squared in the first term and squared again in the second term. \(Var(cX) = c^2E[X^2] - c^2(E[X])^2\)

Factor out the constant from the entire expression

Now, we can factor out the constant squared, \(c^2\), from the entire expression to obtain the desired result. \(Var(cX) = c^2(E[X^2] - (E[X])^2) = c^2 Var(X)\) Now, let's prove property (b), \(Var(c+X) = Var(X)\):

Use the definition of variance with c+X

We will replace X with c+X in the definition of variance to find the variance of c+X. \(Var(c+X) = E[((c+X))^2] - (E[c+X])^2\)

Expand the squared term and apply the linearity of expectation

We will first expand the squared term and then apply the linearity of expectation to both \(E[((c+X))^2]\) and \(E[c+X]\). \(Var(c+X) = E[c^2 + 2cX + X^2] - (c + E[X])^2\)

Apply the linearity of expectation to each term

We will now apply the linearity of expectation to each term inside the expectations. \(Var(c+X) = E[c^2] + 2cE[X] + E[X^2] - (c^2 + 2cE[X] + (E[X])^2)\)

Simplify the expression by cancelling out equivalent terms

We will now simplify the expression by cancelling out equivalent terms, which will simplify to the desired result. \(Var(c+X) = E[X^2] - (E[X])^2 = Var(X)\) We have successfully proven both properties (a) and (b).