Description

1. Consider the function
, −1 ≤ x ≤ 1
fX otherwise
(a) Find c for fX(x) to be a valid density function.
(b) Let X be a random variable with density function fX(x). Find the distribution function of X.
(c) Find the mean of X.
(d) Find the variance of X.
(e) Find the expected value of (X −1)2.
2. A professor offers an exam for which a randomly chosen student’s grade can be modeled as a Gaussian random variable, X. The following grading scheme is used:
• if X ≥ 85, the grade is A
• if 70 ≤ X < 85, the grade is B
• if X < 70, the grade is C
(a) If the mean exam score is 82 and the variance is 64, find the probabilities of A, B, and C.
(b) If a student is confident that he made at least an 80 on the exam, what is the probability that the student made an A on the exam?
(c) Suppose that the professor instead wants to adjust the difficulty of the problems of the exam so that the probability of an A is 0.25 and the probability of a B is 0.52. What should the values of the mean and variance be to achieve these probabilities?
3. Four boxes of computer chips arrive. Each box is from a different manufacturer. The boxes contain, respectively, 250, 300, 400, and 500 chips. One of the boxes is chosen at random. Let X denote the total number of chips in the chosen box. The boxes are combined, and one of the chips is selected at random. Let Y denote the number of chips that were in the box from the selected chip.
(a) Which of E[X] or E[Y] do you think is larger? Why?
(b) Compute E[X] and E[Y].
4. The Laser Interferometer Gravitational-Wave Observatory (LIGO) is very sensitive – it is designed to detect a change in distance between its mirrors 1/10,000th the width of a proton! Thus, it also detects very minor disturbances, such as a car driving by, or someone walking down the hall. On average, the LIGO inferometer at one of the two locations detects an average of 30 disturbances per hour that are not gravitational waves.
(a) What is the probability that there will be at least one disturbance in a 1 minute period? (b) What is the probability that there will be at least one disturbance in a 5 minute period?
(c) What is the probability that there will be fewer than 5 disturbances in a 15 minute period?
(d) What is the probability that there will be more than 10 disturbances in a 15 minute period?
(a) at least 50 who are in favor of the proposition; (b) between 60 and 70 who are in favor of the proposition; (c) fewer than 75 in favor.
6. Samples from a certain low noise amplifier are distributed Gaussian with mean 0 and variance 1. Samples from the same company’s power amplifier are Gaussian with mean 0 and variance 4. During a particular day, the chip labeler breaks down, but nobody noticed until chips of both types have been mixed together.
Recall that by the “weighted density functions”, we refer to the numerator of the a posteriori probabilities, which is the product of the likelihoods and the a priori probabilities.
This problem requires both plotting work (in a Jupyter notebook) and analytical work (that I recommend you do by hand on paper).
(a) First consider that chips of both types are equally likely. Plot the weighted density functions in Jupyter notebook. Vary the regions plotted to zoom in on any points necessary to give a reasonable estimate (at least accurated to within ±0.25) of the MAP decision regions.
(b) The MAP decision regions in this problem are determined by a set of thresholds. (Unlike the case covered in class, there is not just a single threshold for each value of the a priori probabilities.) Give a formula for the MAP decision thresholds
(c) Write a function in your Jupyter notebook that inputs the probability that a randomly chosen chip is a low noise amplifier. The function should
• plot (using plt.plot) both the weighted density functions
• calculate the numerical value of the MAP decision thresholds
• plot (using plt.scatter) the points where the x-values is a MAP decision threshold and the y-value is the value of the weighted densities at that MAP decision threshold
(d) Show the output of your function when there are equal numbers of each type of amplifier
(e) Show the output of your function when there are twice as many power amplifiers as low-noise amplifiers
(f) Show the output of your function when the probability of a randomly chosen chip being a low-noise amplifier is 0.9
(g) What happens when the probability of a random chosen chip being a low-noise amplifier is 0.2?
(h) Plot the weighted density functions using a logarithmic y-axis (using plt.semilogy) when the probability of a random chosen chip being a low-noise amplifier is 0.2
(i) Show where the math breaks down when the probability of a random chosen chip being a low-noise amplifier is low