Test P

Question 1

Cov(X,X) = ?

Answer 1

Var(X)

Question 2

What equation relates Var(X,Y) with covariance?

Answer 2

Var(X,Y) = Var(X) + Var(Y) + 2Cov(X,Y)

Question 3

Shortcut formula for Var(X)?

Answer 3

Var(X) = E(X²) - E(X)²

Question 4

Var(aX) = ?

Answer 4

a²Var(X)

Question 5

Cov(aX₁ + bX₂, Y₁ + Y₂) = ?

Answer 5

aCov(X₁,Y₁) + aCov(X₁,Y₂) + bCov(X₂,Y₁) + bCov(X₂,Y₂)

Question 6

If X and Y are independent, then M_X+Y(t) = ?

Answer 6

M_X(t) x M_Y(t)

Question 7

If X and Y are independent, then Var(X+Y) = ?

Answer 7

Var(X) + Var(Y)

Question 8

f_Y|X=x(y|X=x) = ?

Answer 8

f_XY(x,y)/f_X(x)

plug in x

Question 9

correlation coefficient p?

Answer 9

p(x,y) = Cov(X,Y)/[√Var(X)√Var(Y)]

Question 10

Cov(X,Y) = ?

Answer 10

E(XY) - E(X)E(Y)

Question 11

Given random variables Y₁ , … , Y_n with common CDF F_Y(y),

let X = max(Y₁ , … , Y_n).

What would F_X(x) equal?

Answer 11

(F_Y(x))ⁿ

Question 12

relationship between f(y) and F(y)?

Answer 12

f(y) = F’(y)

Question 13

E(X) = ?

X is continuous with PDF f_X(x)

Answer 13

E(X) = ∫x f_x(x) dx

Question 14

Bayes Theorem:

Pr(U|D) = ?

where S,P,U are a partition of probability space

Answer 14

Pr(D|U)Pr(U)

____________________________

Pr(D|S)Pr(S) + Pr(D|P)Pr(P) + Pr(D|U)P(U)

Question 15

Necessary condition for independence of two continuous random variables:

region where joint density is positive is _________, and if joint density is product of _______.

What can you conclude about Cov(X,Y) if X and Y meet these conditions?

Answer 15

rectangular with sides parallel to axe

function of first variable only and function of second variable only

Cov(X,Y) = 0

Question 16

Variance of uniform distribution from (a,b)?

Answer 16

(b-a)²/12

Question 17

Darth Vader Rule

E(max(T,2)) = ?

Answer 17

2 + ∫₂^∞ s_T(x) dx

Question 18

Survival function of exponential random variable T with mean 3?

Answer 18

s_T(t) = e^-t/3

Question 19

Survival function in terms of F?

Answer 19

s(x) = 1 - F(x)

Question 20

Darth Vader Rule

E(X) = ? in terms of s(x)

Answer 20

E(X) = ∫₀^∞s(x) dx

Question 21

Let’s say F(x) = 0 for 0 < x < 1

and F(x) = (1-(1/y³))³ for x > 1

Describe s(x)

Answer 21

s(x) = 1 for 0 < x < 1

s(x) = 1 - (1/y³))³

Question 22

If events A and B are independent, what 3 things are true:

Answer 22

Pr(A∩B) = Pr(A)Pr(B)

Pr(A|B) = Pr(A)

Pr(B|A) = Pr(B)

Question 23

if X = g(Y) and f_x is PDF of X

then f_Y(y) = ?

Answer 23

f_X(x(y)) |dx/dy|

Question 24

∫₀^∞ yme^-my dy = ?

Answer 24

Mean of exponential random variable with hazard rate m;

1/m

Question 25

PDF of Poisson (Pr(N=n)) with lambda = L

Answer 25

f_N(n) = (e^-LLⁿ)/n!

Question 26

Pr(A∪B) = ?

Answer 26

Pr(A) + Pr(B) - Pr(A∩B)

Question 27

X and Y are bivariate normally distributed with both means μ_x = μ_y = 0, σ_x², σ_y² and correlation p_xy. What is Var(Y|X=x)?

Answer 27

Var(Y|X=x) = (1-p_xy²)(σ_y²)

Question 28

Var(aX+bY+cZ) when given Cov between X Y and Z

Answer 28

a²Var(X) + b²Var(Y) + c²Var(Z) + 2abCov(X,Y) + 2acCov(X,Z) + 2bcCov(Y,Z)

Question 29

Recall this key formula which relates Var(Y) to Var(Y|X)

Answer 29

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

Question 30

ChebyShevs Inequality?

Answer 30

Pr(|X-u|/σ _\>_ r) _\<_ 1/r²

Question 32

X has CDF F_X(x). Let X₍₁₎, ..., X_(n) be ordered statistics. What is F_X(n)(x) and S_X(n)(x)?

Answer 32

F_X(n)(x) = (F_X(x))ⁿ S_X(n)(x) = 1 - (F_X(x))ⁿ

Question 33

X has CDF F_X(x). Let X(1), ..., X(n) be ordered statistics. What is F_X(1)(x) and S_X(1)(x)?

Answer 33

F_X(1)(x) = 1 - (1 - F_X(x))ⁿ S_X(1)(x) = (1 - F_X(x))ⁿ ... i think check NB

Question 35

What does the hazard rate function equal in terms of the survival function s(x)?

Answer 35

d/dx -(ln(s(x)))

Question 36

How to find E(X²|Y=y) when given fxy?

Answer 36

Integrate x² f_x(x|Y=y)

Question 37

What is the PDF of an exponential distribution with hazard rate a?

Answer 37

ae^-ax

Question 38

Moment generating function of exponential function with hazard rate a?

Answer 38

M_T(t) = a/(a-t)

Question 40

convolution formula: let Z = X + Y, what is f_X+Y(z) in terms of f_XY?

Answer 40

Integral of f_XY(x,z-x) dx