# Carroll College Exponential Random Variable Questions

Description

I need an answer for questions 1, and 2 only. *The sample is attached and HWs solution is attached. in this link https://www.filefactory.com/file/4y6pkt9yg5xm/Clas…Hint: When you open the class notes, you will find some explanations which have been written to me, but you can jump to the examples or search by keywords in the pdf file.* if you need more class notes, let me know. < But some of them are related to the previous questions.We are in very limited time so the time period to do these questions is 5 hours. The average time to solve the two questions is 1 hour and it might take the mathematician 0.5 hours to do them ALL < I doubled the time to make you do the questions with confidence 11 attachmentsSlide 1 of 11attachment_1attachment_1attachment_2attachment_2attachment_3attachment_3attachment_4attachment_4attachment_5attachment_5attachment_6attachment_6attachment_7attachment_7attachment_8attachment_8attachment_9attachment_9attachment_10attachment_10attachment_11attachment_11 Unformatted Attachment Preview 711 Test 2 (2021 Spring) Problem 1 (25 points) x1 , x2 , . . . , xn are i.i.d. samples of an exponential random variable X with probability density function (pdf) ( p(x) = βe−βx , if x ≥ 0 0, otherwise (1) where β > 0 is a parameter.
a) Find the maximum-likelihood estimate (MLE) of β.
b) Find the maximum a posteriori (MAP) estimate of β if we have a prior for β, given by
(
p(β) =
(1/a)e−β/a , if β ≥ 0
0,
otherwise
(2)
where a > 0 is a constant.
Problem 2 (25 points)
Suppose the joint pdf of random variables X and Y is
(
p(x, y) =
c, if (x, y) ∈ A
0, otherwise
(3)
where c > 0 is a constant and A is the “triangular” region given in Fig 1, with two straight line
boundaries (x = 0 and y = 0) and one curved boundary (y = 1 − x2 ). Suppose we want to find a
linear estimate of Y from X, with Yb = w1 X + w0 , where w1 and w0 are weights.
a) Find the weights w1 and w0 that will minimize the mean square error E[(Y − Yb )2 ].
b) What is the minimum mean square error achieved by the w1 and w0 you have found in part
a)?
Figure 1: Region A for Problem 2.
Problem 3 (25 points)
For Problem 2, suppose we want to use a conditional expectation estimator to estimate Y from X.
a) Find the conditional expectation estimator of Y , given by Yb = E[Y |X].
b) Find the mean square error of the conditional expectation estimator. Compare this with the
result of Problem 2 part b), what can you conclude?
Problem 4 (25 points)
a) In Fig 2, there are a triangular region A and a square region B on the plane. We can define a
function f (x1 , x2 ) on the entire plane as
(
f (x1 , x2 ) =
1, (x1 , x2 ) ∈ A ∪ B
0, otherwise
(4)
where A ∪ B denotes “A or B.” Can f (x1 , x2 ) be implemented with a neural network with
no more than 4 layers (i.e., an input layer, no more than two hidden layers, and an output
layer)? If yes, show how and the neural network weights (you can use φ(t) as the non-linear

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