Homework 18: Law of the Unconscious Statistician

1. Let \(X\) have pdf $$ f_X(x)=2(1-x), 0\leq x\leq 1 $$ Suppose that \(Y=X^2\). Find \(E(Y)\) in two different ways.
2. A box is to be constructed so that its height is five inches and its base is \(X\) inches by \(X\) inches, where \(X\) is a random variable with pdf \(f_X(x)=6x(1-x)\), \(0\lt x\lt 1\). Let \(V\) be the volume of the box. Find \(E(V)\).
3. Suppose that the hypotenuse of an isosceles right triangle is a random variable which is uniform over the interval \([4,10]\). Find the expected value of the area of the triangle.
4. Grades on a recent math exam were fairly low. When graphed, the distribution of grades had a shape similar to the pdf $$ f_X(x)=\frac{1}{5000}(100-x), \ 0\leq x \leq 100 $$ In order to ``curve'' the results, the professor announces that she will replace each grade \(X\) with a new grade \(g(X)\), where \(g(X)=10\sqrt{X}\). Will this curve raise the class average above 60?
5. Suppose that the lifetime (in years) of a light bulb is a random variable \(X\) with pdf \(f_X(x)=0.5e^{-0.5x}\), \(x\geq 0\). Alice buys a new bulb today, but when it burns out, she will have to buy another one. The price of a light bulb is currently \(3\) dollars. Due to inflation, the price \(t\) years from now will be \(3e^{0.1 t}\) dollars. Find the expected price of the second bulb.