STOCHASTIC
MODELS
1
RECITATION 2
1. Test two integrated circuits one after the other. On each test, the possible outcomes
are a, (accept) and r (reject). Assume that all circuits are acceptable with probability
0. 9 and that the outcomes of successive tests are independent. Count the number of
acceptable circuits X and count the number of successful tests Y before you observe
the first reject. ( If both tests are successful, let ·y = 2.)
a) Draw a tree diagram for the experiment and find the joint PMF PX,Y(x, y) .
b) Find the marginal PMFs for the random variables X and Y .
2. Random variables X and Y have joint
a) Find the constant c and P [A] = P[2 < X < 3, 1 < Y < 3].
b) What is P [A] = P[Y > X]?
3. The joint probability density function of random variables X and ·Y is
Find the constant c. What is the probability of the event A = X^2 + Y^2 < 1?
4. The joint PDF of X and ·Y is
Find the marginal PDFs f x(x) and f y(y).
5. The joint density for the random variables (X, Y ), where X is the unit temperature change
and Y is the proportion of spectrum shift that a certain atomic particle produces, is
(a) Find the marginal densities fx(x), fy(y), and the conditional density f(y|x).
(b) Find the probability that the spectrum shifts more than half of the total observations, given
that the temperature is increased by 0.25 unit.
