K:\LintonT\classes\spring00\Statistics\diesel.html

Diesel Engine Start Times

Mathematica notebook

Vehicles with diesel engines need to be warmed up each morning before being started. Let X denote the ambient temperature in degrees Centigrate, and therefore, the initial temperature of the engine. Let Y denote the time, in minutes, required to warm up the engine before starting the vehicle. Let S denote the set of pairs (x,y) with 0 <= x <= 40 and 0 <= y <= 2. Assume that the joint probability density function of X and Y is given by

for (x,y) in S, and f[x,y] = 0 otherwise. Double click on the dark triangles to see Tom's answers (compliments of Mathematica).

Find the probability that on a random morning the ambient temperature exceeds 20 degrees and that it takes at least one minute to start the vehicle, or P[X > 20 and Y > 1].

We integrate the density function over the approriate region:

Find the marginal density functions of X and Y.

Find the probability that on a random morning the ambient temperature exceeds 20 degrees.

Find the probability that on a random morning it takes more than 1 minute to start the vehicle.

Are X and Y independent?

Based on the last two answers and the first probability, P[X > 20 and Y > 1] = 0.373494, it seems possible (since 0.740964 * 0.506024 is almost equal to 0.373494), but if we check whether or not fx[x]*fy[y] = f[x,y], we see that:

the product of the marginal density functions has an x*y term that is NOT present in the joint density function, so X and Y are NOT independent.

Find E[X], E[Y], sx and sy.

Before computing the covariance of X and Y, COV[X,Y], do you expect COV[X,Y] to be positive or negative? Why?

Negative, as X = temperature goes up, the time needed to warm up the engine should go down, that is, above average values of X should be more likely with below average values of Y.

Calculate, both directly and with the shortcut formula, COV[X,Y].

Note, this result (a tiny negative covariance) says that the ambient temperature has very little effect on the starting time of a diesel engine!

One way to measure the impact that various values of X have on Y is to calculate the expected value of Y, given X = a (and vary the value of a, from 0 to 40). If you think about it some, if we know X = 20 and we want to calculate the expected value of Y in this setting, it seems reasonable to integrate y times the conditional density of "Y given X = 20" from y = 0 to y = 2. Now, if we wanted to calculate the expected value of Y given X = 10, we'd simply integrate y times the conditional density of "Y given X = 10" (over all values of Y). Thus, we might look for a function, say effect[a], which on input x = a, gave us the expected value of Y, given X = a. What we've seen in the above calculation of COV(X,Y), is that this function will probably be decreasing, but it won't likely decrease very much (the covariance was a tiny negative number). Your assignment is to come up with a plot of the function effect[a] (from a = 0 to 40) and explain how you came up with your plot, and why you think it is correct.