Problem: expected value and variance of a linear function of a random variable

Problem

• Suppose that \(X\) is a discrete random variable. Show that \(E\left( a+bX \right)=a+bE\left( X \right)\) . Hint: Let \(Y=a+bX\) and use the general formula

\[E\left( Y \right)=\sum_{i=1}^{n}{ g\left( x_i \right)f\left( x_i \right) }\]

• Suppose that \(X\) is a discrete random variable. Show that \(Var\left( a+bX \right)=b^2Var(X)\) . Hint: Let \(Y=a+bX\) then \(Var\left( Y \right)=E{\left( Y-μ_Y \right)}^2\) (see xx) Show that \({\left( Y-μ_Y \right)}^2=b^2{\left( X-μ_x \right)}^2\) . Finally, use the result in a)
• Same as a) but \(X\) is a continuous random variable.