The p.d.f. of a random variable X is f(x) = 1 - x/2 for 0 ≤ x ≤ 2. Suppose that measurements are very imprecise, and that all values of X ≤ 1 are recorded as 0.5 and all values of X > 1 are recorded as 1.5. Write a random variable describing these imprecise measurements, find the associated probabilities, and compute the expectation.
For each continuous random variables with the given p.d.f, find a discrete random variable that approximates it. Graph the histogram for this discrete random variable and compare it with the p.d.f. for the continuous random variable. Find the expectation of the discrete random variable, and check whether it is equal to the expectation of the continuous random variable.