Himalaya (Exam 2000, Question 1

Usage notes

a: Probability density functions

The heights of Mt Everest and Mt K2 have been measured three times: in 1994, in 1998 and with GPS sensors. The collected data is given by:

		Mt Everest		Mt K2
	1994	8847.5 +/- 0.3	8609.0 +/- 0.5
	1998	8848.0 +/- 0.2	8611.0 +/- 0.2
	GPS	8848.31 +/- 1	8611.50 +/- 1

Assume that the 6 measurements given above are realizations of independent, normally distributed random variables. Give a probability density function for the random variable H_Everest,94 (the result of the random experiment of measuring the height of Mt Everest using the method applied in 1994). Note that this probability density function will depend on h_Everest, the real height of Mt Everest, which you can simply write as
hEverest in the input-field.

f₉₄(u | h_Everest) = ___________________ for all real u.

What is the joint density of the random variables H_Everest,94, H_Everest,98 and H_Everest,GPS?

f(u, v, w | h_Everest) = ___________________ for all real u, v, w.

b: Maximum Likelihood Revisited

Suppose that random Variables X₁...X_n have a joint density or frequency function f(x₁,x₂,x₃,...,x_n|Theta) depending on some parameter Theta. Given observed values for the x_i (i=1...n), the likelihood of Theta as a function of x₁,x₂,...,x_n is defined as:

lik(Theta) = f(x₁,x₂,...,x_n|Theta)

The maximum likelihood estimate (mle) of Theta is that value of Theta that maximizes the likelihood - that is, makes the observed data "most probable" or "most likely". (Compare: "Mathematical Statistics and Data Analysis" by John A. Rice)

c: Estimating the height of Mt Everest

Now we want to estimate the height of Mt Everest. The maximum likelihood estimate of the parameter hEverest is that value of hEverest which makes the observed data most likely, i.e. which maximizes the likelihood of hEverest. What is the likelihood of hEverest given the above measurement data?

lik(h_Everest) = ___________________

To simplify the problem of maximizing lik(h_Everest), find the simplest expression g(h_Everest) which is minimal exactly if lik(h_Everest) is maximal.

g(h_Everest) = ___________________

d: The least squares estimate for the heights of Mt Everest and Mt K2

Write down the equation which you have to solve in order to get a least squares estimate of h_Everest.

___________________

Now, using your previous result, what is the most likely height of Mt Everest (round to 2 digits after the decimal point)?

___________________ m

Estimate also the height of Mt K2.

___________________ m

e: Additional information

Now assume that h_K and h_E have been measuered simultaneously with GPS. It is known that the difference between simultaneous GPS measures has a much higher accuracy, i.e., h_E-h_K=236.81+-0.01 . (h_E=h_Everest...)

The collected data and the additional information give us an equation system with:
___________________ equations and ___________________ unknowns.

Write down the sum of square errors we want to minimize. Use the above defined e_i in your expression.
S2 := ___________________ ;

What is the solution of this new least square problem ? Write down the estimations for h_E and h_K.
hE = ___________________ m. hK = ___________________ m.

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