4. Suppose you are required to predict the value of the r.v. X. Assume that you know the distribution of X as well as the mean µ and variance σ². Suppose your prediction is a real number 0. The squared error (X – 0)² will be deducted from your paycheck. - (a) If your prediction is 0 = 2 and X turns out to be 7, what is your squared error? (b) If your prediction is 0 = 3/4 and X has a Bernoulli distribution with mean 1/3, what is your mean squared error? That is, compute E[(X - 0)²]. (c) If X has a uniform-(0, 1) distribution and your prediction is 0 = 1/3, what is your mean squared error?
4. Suppose you are required to predict the value of the r.v. X. Assume that you know the distribution of X as well as the mean µ and variance σ². Suppose your prediction is a real number 0. The squared error (X – 0)² will be deducted from your paycheck. - (a) If your prediction is 0 = 2 and X turns out to be 7, what is your squared error? (b) If your prediction is 0 = 3/4 and X has a Bernoulli distribution with mean 1/3, what is your mean squared error? That is, compute E[(X - 0)²]. (c) If X has a uniform-(0, 1) distribution and your prediction is 0 = 1/3, what is your mean squared error?
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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![4. Suppose you are required to predict the value of the r.v. X. Assume that you
know the distribution of X as well as the mean µ and variance o². Suppose
your prediction is a real number 0. The squared error (X - 0)² will be
deducted from your paycheck.
(a) If your prediction is 0 = 2 and X turns out to be 7, what is
error?
your squared
(b) If your prediction is 0 = 3/4 and X has a Bernoulli distribution with
mean 1/3, what is your mean squared error? That is, compute
E[(X - 0)²].
(c) If X has a uniform-(0, 1) distribution and your prediction is 0 = 1/3,
what is your mean squared error?
(d) In general, if you wish to minimize your mean squared error
E[(X - 0)²]
What should
your prediction 0 be?
(Hint: There are a variety of ways to solve this, but one way is to start
by
E[(X - 0)²] = E[((X − µ) + (µ − 0))²1
-
and then just continue doing what seems like the obvious next step until
you end up with a nice expression for the mean squared error. In this
expression it should be easy to see how to pick to minimize the mean
squared error.)
(e) Based on the previous part, can you come up with a nice expression for
the mean squared error if your prediction of X is 0?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb9e27b5d-0ab8-428e-954d-97d64fc14c61%2F72930599-cd98-4211-9e18-20dbb773ddc9%2F8hc5lcq_processed.png&w=3840&q=75)
Transcribed Image Text:4. Suppose you are required to predict the value of the r.v. X. Assume that you
know the distribution of X as well as the mean µ and variance o². Suppose
your prediction is a real number 0. The squared error (X - 0)² will be
deducted from your paycheck.
(a) If your prediction is 0 = 2 and X turns out to be 7, what is
error?
your squared
(b) If your prediction is 0 = 3/4 and X has a Bernoulli distribution with
mean 1/3, what is your mean squared error? That is, compute
E[(X - 0)²].
(c) If X has a uniform-(0, 1) distribution and your prediction is 0 = 1/3,
what is your mean squared error?
(d) In general, if you wish to minimize your mean squared error
E[(X - 0)²]
What should
your prediction 0 be?
(Hint: There are a variety of ways to solve this, but one way is to start
by
E[(X - 0)²] = E[((X − µ) + (µ − 0))²1
-
and then just continue doing what seems like the obvious next step until
you end up with a nice expression for the mean squared error. In this
expression it should be easy to see how to pick to minimize the mean
squared error.)
(e) Based on the previous part, can you come up with a nice expression for
the mean squared error if your prediction of X is 0?
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