Level 3

Question 1

Given joint pdf f(x, y), what is
P(X > Y)?

Answer 1

Just use any 1 variable to show the inequality not both like:
P(X > Y) = ∫{-∞}^{∞} ∫{y}^{∞} f(x, y) dx dy

P(X > Y) = ∫{-∞}^{∞} ∫{-∞}^{x} f(x, y) dy dx

Question 2

MGF of sum of independent r.v (Sn)

Answer 2

M_{X+Y}(t) = M_X(t) * M_Y(t)

Question 3

Gamma function and value

Answer 3

Γ(n) = ∫_{0}^{∞} e^{-x} x^{n-1} dx
Γ(n) = (n - 1)!

∫_{0}^{∞} e^{-x} x^{n} dx = n!

Question 4

Indicator variable trick use

Answer 4

X = Sum (Ii)
E[X] = Sum E[Ii]
E[X] = Sum P(favourable)

Question 5

Coupon collector problem steps (# of Dice rolls to get all faces at least once)?

Answer 5

1) T = T1 + T2 +…+ T6
2) P(T1)=1, P(T2)=5/6, P(T3)=4/6…
3) E[T1]=1/1, E[T2]= 1/(5/6)…
4) E[T] = E[T1] + E[T2] +…+ E[T6]
= 6(1/1+1/2+…+1/6) = 14.7 rolls
Hn = ln(N) + Euler Mascheroni
Euler Mascheroni = 0.577

Question 6

Linear transformation of r.v.
What is X = a + bY, where
Y ~ N(μ, σ²)?

Answer 6

X ~ N(a + bμ, b²σ²)

Question 7

Law of total variance: Var(X) using conditional expectation and conditional variance

Answer 7

Var(X) = E[Var(X|Y)] + Var(E[X|Y])

Question 8

Law of total expectation: E[X] using conditional

Answer 8

E[X] = E[E[X|Y]]

Question 9

All odd and even moments of Normal Distribution values.

Answer 9

Odd moments = 0
E[X^(2n)] = (2n-1) !!
Where (2n - 1)!! = 1.3.5.7….(2n-1)

Question 10

Ws, Wt ~ Brownian Motion
What is Cov(Ws, Wt)

Answer 10

min(s, t)

Question 11

Affine function meaning.

Answer 11

A linear function plus a constant shift, not passing necessarily through the origin.

Eg: f(x) = 6x + 3 -> affine

An affine function in x: (a number times x plus another number).