Retake Exam — FINAL
Probability and Statistics
Ingenierı́a Biomédica - INGLÉS
29 of June 2021
Please write neatly. Answers which are illegible for the grader cannot be given credit. Define
the random variables that you are going to use, mention the theorems and the definitions that you
are using and justify the steps followed. You have 180 minutes time to complete your work. You
are allowed to use a calculator and two sheets with annotations. Question 1 (f ) is OPTIONAL.
Problems
1. The pair (X, Y ) has joint cdf given by:
(
FX,Y (x, y) =
c(1 − 1/x2 )(1 − 1/y 2 )
0
for x > 1, y > 1
elsewhere
(a) (0.25 points) Find c.
(b) (0.75 points) Find the marginal cdf and the marginal pdf of X and Y .
(c) (0.75 points) Find the probability of the following events: {X < 3, Y ≤ 5} and
{X > 4, Y > 3}.
(d) (0.25 points) Find the probability of the rectangle R = [0, 2] × [0, 2].
(e) (1 point) Find E[XY ].
(f) (+0.5 points) Find the correlation between X and Y .
2. In a video game, the number of times I get eliminated follows a Poisson distribution. Assuming that on average I get eliminated 4 times per game and that the result between games is
independent, estimate:
(a) (1 point) The probability of being eliminated more than 370 times in a season, if a
season consists of 100 games.
(b) (1 point) The probability of being eliminated between 100 and 120 times in a tournament of 25 games.
3. Let X(t) be a continuous-time Gaussian random process with mean function and covariance
function given by:
mX (t) = 3t CX (t, s) = 9e−2|t−s| .
(a) (0.25 points) Is the process wide-sense stationary?
(b) (0.75 points) Compute the correlation coefficient function. Make a sketch of the
function using τ = t − s as the abscissa variable. Are X(1) and X(100) independent?
(c) (0.5 points) Find the mean and variance of Y = X(1) + X(2).
(d) (0.5 points) Find P [X(3) < 9] and P [X(1) + X(2) > 15].
1
4. Consider a random process X(t) defined by
X(t) = A cos t + B sin t,
−∞ < t < ∞
where A and B are independent random variables with distributions: A ∼ U (−3, 3) and
B ∼ N (0, σ 2 = 3).
(a) (0.5 points) Compute E[A], E[B], E[A2 ], E[B 2 ].
(b) (0.75 point) Compute the mean and variance functions of the process X(t).
(c) (1.5 point) Compute the autocorrelation and autocovariance functions. Is the process
wide-sense stationary?
(d) (0.25 points) Are the random variables X(0) and X(π) correlated?
Observation:
cos (α + β) + cos (α − β)
2
cos (α − β) − cos (α + β)
sin α sin β =
2
cos α cos β =
2
