1. A random variable X takes values between -2 and 3 with a probability density function 15 f(x) = -2 sxs0 64 64 3 f(x) =+ cx. and f(x) = 0 clsewhere. (a) Find the value of c and sketch the probability density function. (b) What is P(-1 sXS 1)? (for "a", do not include the sketch part) 2. (Answer items "a" without the sketch part; and "c") A car panel is spray-painted by a machine, and the technicians are particularly interested in the thickness of the resulting paint layer. Suppose that the random variable X measures the thickness of the paint in millimeters at a randomly chosen point on a randomly chosen car panel, and that X takes values between 0.125 and 0.5 mm with a probability density function of f(x) = A(0.5 - (x- 0.25)) for 0.125 < x < 0.5 and f(x) = 0 elsewhere. (a) Find the value of A and sketch the probability density function. (b) Construct and sketch the cumulative distribution function. (c) What is the probability that the paint thickness at a particular point is less than 0.2 mm? A random variable has a probability density function (x) = A(x- 1.5) over the state space 2 s* < 3. (a) What is the value of A? (b) What is the median of the random variable? (c) What is the standard deviation of the random variable? 3.