Description

1. Consider the function
(a) Find c for fx(x) to be a valid
(b) Let X be a random variable with 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 2 85, the grade is A
• if 70 SX < 85, the grade is B
• if X < 70, the grade is C
(a) If the mean exam scor is 82 and the variance is 64, find the p obabilities of A, B, and
(b) If a student is confident that he made at least an 80 on the am, what is the ro ability that the student made an A on the exam? o.asq O S9
(c) Suppose that the professor instead wa teadjust the di culty of the prob em th exam so that the probability of an A is 0.25 and the probability of a B is 0.523-What should the values Of the mean and variance be to achieve these probabilities?
3. Four boxes of co contain, respectively, 2 O;-400öandSOOochips. One of the boxes is chosen at random. Let X denote the total number of chips in the chosen box. The boxes are of the chips is selected at random. Let Y denote thenumber of chips that were in the box from the selected chip.
(a) Which ofE[XJ or ELY] do you thin is larger? Why?
(b) Compute E[X] and E[Y]. bbl
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4. The Laser Inte erometer Gravitationa designed to detect a change in distance betwe i mirrors 1/10,000th the width of a proton! Thus, it also detects very minor disturbances, such as ac driving by, or someone walking down the hall. On average, the LIGO inferometer at one of the two locatiOnsZetects -an average of 30 disturbances per hour that are not gravitational waves.
(d) What is the there will be more than 10 disturbances in a 15 minute peri d?
(a) at least 50 who are in favor of the proposition;
(b) between 60 and 70 who are in favor of the roposition;
(c) fewer than 75 in favor.
6. Samples from a certain low noise amplifier are distributed Gaussian with mean Q and variance 1.
variance
Samples from the same company’s power amplifier are Gaussian with mean 0 and
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).
V/ (a) First consider that chips of both types are equally likely. Plot tfweighted density functions in Jupyter notebook. Vary the regions plotted to zoom i on any points necessary to give a reasonable estimate (at least accurated to within±d.25) of the-M& decision r ions
(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 MAPdecision 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
J (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. semi logy) 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
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