**Name** (please print):
___________________________________________________________________________________

1. *What is cyberinfrastructure.*

Explain what is cyberinfrastructure, why it is useful, and why it did not appear
many years ago, when computer communications were slower.

2. *Statistical foundations.*

Explain, in detail, why for two independent events A and B, the probability
of A & B is equal to the product of the probabilities of A and B: P(A & B) =
P(A) * P(B).

3. *Statistical foundations.*

What is the Maximum Likelihood Method?
Explain, in detail,
how for the Gaussian distribution, with the probability density
d(x)= const * exp(-(1/2)*(x-a)^{2}/s^{2}), this method leads to
an estimate of the mean a as the arithmetic average of the measurement results
x_{1}, ..., x_{n}:
a = (1/n) * (x_{1} + ... + x_{n}).

4. *Linearization.*

Illustrate, on the example of the function
y = x_{1}^{2} + x_{2},
with the nominal
values x_{1} = 1.0 and x_{2} = 2.0 and measurement errors
dx_{1} = 0.1 and dx_{2} = -0.1,
what will be the error dy as estimated by the linearization method, and how
this estimated error compares with the actual value of this error. Compare
the analytical expressions for the corresponding partial derivatives with the
results of numerical differentiation.

5. *Data fusion.*

Let us assume that we have measured the same quantity with
two different measurement instruments. Both measuring instrument have 0 mean;
the first instrument has standard deviation 0.1, the second has standard deviation
0.2. The result of the first measurement is 1.1, the result of the second
measurement is 0.9. Combine these two results into a single "fused" value. What
is the accuracy (i.e., in this case, what is the standard deviation) of this fused
value? Where do the resulting formulas come from (no need for detailed derivations,
just explain the main ideas.)

6. *Need for data processing.*

Explain why we cannot directly measure
all physical quantities of interest to us, why we need data processing. Feel free
top use either a radar example that we discussed in class or any other example.

7. *Data processing: uncertainty.*

Let us assume that data
processing is performed by using
a function
y = f(x_{1}, x_{2}) = x_{1}^{2} + x_{2},
the measurement results are X_{1} = 1.0 and X_{2} = 2.0, and
the measurement errors are independent and normally distributed, with 0 mean
and standard deviations s_{1}=0.1 and s_{1}=0.2. Find the
standard deviation s of the result of data processing.

8. *Uncertainty in data processing: computational aspects.*

For the formula for
computing uncertainty (i.e., standard deviation) of the result of data processing,
explain how the computational complexity (= number of computational steps)
depends on the choice of the
parameters h_{i} used in numerical differentiation, and what is the choice
for which the computational complexity is the smallest.