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You have two instruments with which to measure the height of a tower. If the true height is 100 meters, measurements with the first instrument vary with mean 100 meters and standard deviation 1. 2 meters. Measurements with the second instrument vary with mean 100 meters and standard deviation 0. 85 meter. You make one measurement with each instrument. Your results are X1 for the first and X2 for the second, and are independent. A) To combine two measurements, you might average them, Y=(X1+X2)/2. What are the mean and standard deviation of Y? b) It makes sense to give more weight to the less variable measurement because it is more likely to be close to the truth. Statistical Theory says that to make the standard deviation as small as possible you should weight X2 is very close to half the variance of X1, so X2 should get twice the weight of X1. That is, use W=1/3X1 + 2/3X2 What are the mean and standard deviation of W?

Answer :

The mean of the combination Y is 100  and the standard deviation of Y is 0.7352   .

In the question ,

it is  given that , the true height of the tower is 100 meter ,

for first instrument ;

the mean is E[X₁] = 100m ;

standard deviation is = 1.2 m

for second instrument ,

mean is E[X₂] = 100 m

standard deviation is = 0.85 m

the combination of the two measurement is Y = (X₁ + X₂)/2

So , the mean of the combination Y is

E[Y] = E[(X₁ + X₂)/2]

= (1/2)[ E[X₁] + E[X₂] ]

Substituting the value of mean from above ,

we get ,

= 1/2[100 + 100 ]

= 200/2 = 100

and the standard deviation is

S.D.(Y) is = √Var(Y)

= √Var((X₁ + X₂)/2)

= √(1/2²)[ Var(X₁) + Var(X₂) ]

=  √(1/4)[ 1.2² + 0.85² ]

= √0.5406

= 0.7352

Therefore , the mean is 100 and the standard deviation is 0.7352  .

The given question is incomplete , the complete question is

You have two instruments with which to measure the height of a tower. If the true height is 100 meters, measurements with the first instrument vary with mean 100 meters and standard deviation 1. 2 meters. Measurements with the second instrument vary with mean 100 meters and standard deviation 0. 85 meter. You make one measurement with each instrument. Your results are X₁ for the first and X₂ for the second, and are independent.

To combine two measurements, you might average them, Y=(X₁ + X₂)/2. What are the mean and standard deviation of Y ?

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