Universidad Rey Juan Carlos Escuela Técnica Superior de Ingenierı́a de Telecomunicación Grado en Ingenierı́a Biomédica (inglés) Probability and Statistics Partial Exam 24 of November 2020 The maximum time to take the exam is 2 hours. You are allowed to use a calculator and two sheets with annotations. Problems 1. Rafa Nadal and Marc López won the gold medal at the 2016 Rio Olympics, after beating the duo formed by Romanians Florin Mergea and Horia Tecau in the final (6-2, 3-6 and 6-4). The Spanish couple had 16 break points1 of which they won 25% and the Romanian couple 5 of which they won 40%. (a) (1 point) Calculate the probability of winning a break point. (b) (1 point) Knowing that a break point is won, compute the probability that it was the Spanish couple. 2. The emperor Marco Ulpio Trajano and his close friend and philosopher Dion Coceyo, play the following game: a dice and a coin are tossed. If the coin results in head Trajano must reward Coceyo an amount of aureus (gold coins) equal to the average of 10 and the result of the dice, otherwise Coceyo pays Trajano the geometric mean2 of 10 and the value of the dice. Let the random variable X be the amount of money won (or lost) by Trajano in one game. (a) (0.5 points) Describe the underlying sample space Ω of this random experiment and specify the probabilities of its elementary events. (b) (0.5 points) Show the mapping from Ω to the range of X, ΩX . Find the probability mass function of X. (c) (1 point) Compute the expected value and the standard deviation. Has any character an advantage? Why or why not? 1 In racquet sports such as tennis, the break point refers to when the player who is not serving at that moment has the possibility of scoring the game by winning the next point. It is a crucial moment in matches that on many occasions decide the direction of the set and the match. √ 2 Let a, b be two positive numbers. The geometric mean of them is ab. 1 3. A random variable X has the density function f (x) = c , x ∈ R. x2 + 1 (a) (0.5 points) Find the value of the constant c. (b) (0.5 points) Find the cdf of X. Make a sketch. (c) (1 point) Find P [X = 0], P [0 < X < 1] and P [ 1/3 < X 2 < 1]. (d) (0.75 point) Compute if exists E[X] and V [X]. 4. The incidence of Covid-19 in Spain is in average 18000 cases per day. (a) (0.5 points) Find an upper bound for the probability that in a day there will be more than 25000 cases. (b) (0.5 points) The standard deviation is 500 cases per day. Using the Chebyshev inequality, find the smallest interval of the form Ik = (µ − kσ, µ + kσ) such that: P (µ − kσ ≤ X ≤ µ + kσ) ≥ 0.9 5. To study fetal malformation an innovative medical 5D ultrasound is used. This modern ultrasound is capable of performing a more defined and realistic reconstruction of the fetus. The fact of having high-resolution ultrasounds increases significantly the fetal malformation detection rate. The system performance is sensible to the voltage; if it is very low or high the images aren’t clear. The medical ultrasound system has a transformer that accepts a voltage V and is able to output a voltage Y = aV + 0.5 · aX, where X ∼ N (0, σ 2 = 2) and a is a parameter that can be adjusted. Actually, the system is working with a = 2 and V = 5 volts. (a) (1 point) Find the distribution of Y . Find the probability that Y < 0. (b) (1.25 points) Over time, the transformer loose efficiency and the noise of the system, X, begins to be biased. Concretely, X ∼ N (µ, σ 2 = µ2 + 1), where µ represents the time that has passed in years. Find the numbers of year that must pass such that P (Y > 20) = 0.9. Observations: Z 1 dx = atan(x) x2 + 1 x atan(x) −∞ −π/2 √ − 3 −π/3 −1 −π/4 √ −1/ 3 −π/6 √ 1/ 3 π/6 0 0 1 π/4 √ 3 π/3 ∞ π/2 Table 1: Values of the inverse tangent function. x Φ(x) −2.33 0.01 −1.96 0.025 −1.64 0.05 −1.28 0.1 0 0.5 1.28 0.9 1.64 0.95 1.96 0.975 2.33 0.99 Table 2: Values of the cumulative distribution function a standard Gaussian random variable. 2
