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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 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?

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter13: Probability And Calculus
Section13.2: Expected Value And Variance Of Continuous Random Variables
Problem 10E
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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?
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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