Week 3 Parameter Estimation

Question 1

what is the law of large numbers

Answer 1

theorem the describes the result of performing the same experiment a large number of times

Question 2

what does the law of large number state

Answer 2

the average obtained from a large number of trials should be close to the expected value and tend closer to this values as the number of trials tends to infinity

Question 3

what does the central limit theorem suggest

Answer 3

in some situations when the independent random variables are added, their properly normalised sum tends toward a normal distribution

Question 4

define estimator

Answer 4

precise, accurate procedure to calculate a given quantity based on observed data

Question 5

define estimation

Answer 5

when you draw a conclusion from a data sample and produce a best value for certain quantities

Question 6

how do you find the expectation value of a function a(x) under a certain normalised probability distribution L(x) for continuous and discrete data

Answer 6

continuous you integrate the product of of the two functions wrt x
discrete you find the the sum of the product of the two functions

Question 7

what are the 4 expectation value laws

Answer 7

E[α + X] = α + E[X]
E[αX] = αE[X]
E[X + Y] = E[X] + E[Y]
E[XY] = E[X]E[Y]

Question 8

what are the five expectation value/variance laws

Answer 8

V(X) = E[(X-E[X])^2] = E[X^2] - E[X]^2
V(α+X) = V(X)
V(αX) = α^2V(X)
V(X+Y) = V(X-Y) = V(X) + V(Y)
V(XY) = E[X^2]E[Y^2] - E[X]^2E[Y]^2

Question 9

what are the three criteria for a good estimator

Answer 9

consistent
unbiased
efficient

Question 10

when is an estimator consistent

Answer 10

when it tends to the true value as the number of data values tends to infinity

Question 11

when is an estimator unbiased

Answer 11

when its expectation value is equal to the true value

Question 12

when is an estimator efficient

Answer 12

its variance is small