2. Maths Primer

Question 1

What is constrained optimisation?

Answer 1

Minimising or maximising an objective function subject to equality constraints

Question 2

When do we use the Lagrangian?

Answer 2

When dealing with constrained optimisation

Question 3

Steps of the lagranian

Answer 3

1. Set up the lagrangian
2. Find the partial derivatives for all variables plus lambda
3. Solve the partial derivatives simultaneously to find the values of the variables

Question 4

What is a probability density function?

Answer 4

It is a function which gives the probability of a certain value or range of values being obtained f(x)

Question 5

What is a cumulative distribution function?

Answer 5

A function which gives the probability of a any value below a certain point being obtained F(x)

Question 6

Expected value of continuous variable x

Answer 6

E(x) = the integral of xf(x) for all x

Question 7

What is the expected value of a discrete random variable

Answer 7

E(y) = the sum of all values of yg(y)

Question 8

What is the equation for variance

Answer 8

Var(z)=E(z^2)- E(z)^2

Question 9

What is E(x-b) equal to?

Answer 9

E(x)-b

Question 10

What is E(ax-by) equal to?

Answer 10

aE(x)-bE(y)

Question 11

What is var(x+b) equal to?

Answer 11

Var(x)

Question 12

What is var(ax-by) equal to?

Answer 12

a^2 x var(x)- b^2 x var(y) + 2abcov(x,y)

Question 13

What is the equation for the covariance of x and y

Answer 13

Cov(x,y)= E(xy)-E(x)E(y)

Question 14

What is the proof that x and y are independent?

Answer 14

Prob(x n y) = prob(x) x prob(y)

Question 15

What is a necessary bit not sufficient condition for x and y to be independent?

Answer 15

E(xy) = E(x)E(y)- cov(x,y) = 0