Expectation & Variance Rules

Question 1

What is Expected Value

Answer 1

• The arithmetic mean of a variable

Question 2

What is E(X) in the discrete case

Answer 2

• ∑xp(x) for all x

Question 3

What is E(X) in the continuous case

Answer 3

• ∫xf(x) dx across all real numbers

Question 4

What is the Var(X) equivalent to

Answer 4

• Var(X) = E[(X - μ)^2] = E(X^2) - μ^2 = σ^2

Question 5

What is Var(X) equivalent to in the discrete case

Answer 5

• ∑(x - μ)^2P(x) for all x

Question 6

What is SD(X)

Answer 6

• SD(X) = σ = +sqrt(σ^2)

Question 7

What is E(a)

Answer 7

• a

Question 8

What is E(aX)

Answer 8

• aE(X)

Question 9

What is E(a +- X)

Answer 9

• a +- E(X)

Question 10

What is E(a +- bX)

Answer 10

• (a +- E(X)) * b

Question 11

What is E(X + Y)

Answer 11

• E(X) + E(Y)
• This can extend for more variables

Question 12

What is E(XY) and what condition has to be satisfied

Answer 12

• E(X) * E(Y)
• X and Y have to be independant

Question 13

What is COV(X,Y)

Answer 13

• E[(X - E(X)) * (Y - E(Y))] = E(XY) - E(X) * E(Y)

Question 14

When is COV(X,Y) = 0

Answer 14

• When X and Y are independent
• If COV(X,Y) is 0 it does not neccasarily mean X and Y are indepedent, but independency gurantees COV(X,Y) = 0

Question 15

What is V(a)

Answer 15

• 0
• Constants do not vary

Question 16

What is V(a +- X)

Answer 16

• V(X)
• Adding a constant to a variable does not change its variance

Question 17

What is V(a +- bX) and prove it

Answer 17

• b^2 * V(X)
• V(a +- bX) = V(bX) = E(b^2X^2) - E(bX)^2 = b^2E(X^2) - b^2E(X)^2 = b^2 * V(X)

Question 18

What is V(X +- Y)

Answer 18

• V(X) + V(Y) +- 2COV(X,Y)

Question 19

What is V(X +- Y) when X and Y are independent

Answer 19

• V(X) + V(Y)

Question 20

Prove that V(X) = E(X^2) - E(X)^2

Answer 20

• V(X) = E[(x - μ)^2] = E(x^2 - 2μx + μ^2) = E(X^2) - 2μE(X) + E(μ^2) = E(X^2) - 2μ^2 + μ^2 = E(X^2) - μ^2 = V(X)