Test P Flashcards
Cov(X,X) = ?
Var(X)
What equation relates Var(X,Y) with covariance?
Var(X,Y) = Var(X) + Var(Y) + 2Cov(X,Y)
Shortcut formula for Var(X)?
Var(X) = E(X2) - E(X)2
Var(aX) = ?
a2Var(X)
Cov(aX1 + bX2, Y1 + Y2) = ?
aCov(X1,Y1) + aCov(X1,Y2) + bCov(X2,Y1) + bCov(X2,Y2)
If X and Y are independent, then MX+Y(t) = ?
MX(t) x MY(t)
If X and Y are independent, then Var(X+Y) = ?
Var(X) + Var(Y)
fY|X=x(y|X=x) = ?
fXY(x,y)/fX(x)
plug in x
correlation coefficient p?
p(x,y) = Cov(X,Y)/[√Var(X)√Var(Y)]
Cov(X,Y) = ?
E(XY) - E(X)E(Y)
Given random variables Y1 , … , Yn with common CDF FY(y),
let X = max(Y1 , … , Yn).
What would FX(x) equal?
(FY(x))n
relationship between f(y) and F(y)?
f(y) = F’(y)
E(X) = ?
X is continuous with PDF fX(x)
E(X) = ∫x fx(x) dx
Bayes Theorem:
Pr(U|D) = ?
where S,P,U are a partition of probability space
Pr(D|U)Pr(U)
____________________________
Pr(D|S)Pr(S) + Pr(D|P)Pr(P) + Pr(D|U)P(U)
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?
rectangular with sides parallel to axe
function of first variable only and function of second variable only
Cov(X,Y) = 0
Variance of uniform distribution from (a,b)?
(b-a)2/12
Darth Vader Rule
E(max(T,2)) = ?
2 + ∫2∞ sT(x) dx
Survival function of exponential random variable T with mean 3?
sT(t) = e-t/3
Survival function in terms of F?
s(x) = 1 - F(x)
Darth Vader Rule
E(X) = ? in terms of s(x)
E(X) = ∫0∞s(x) dx
Let’s say F(x) = 0 for 0 < x < 1
and F(x) = (1-(1/y3))3 for x > 1
Describe s(x)
s(x) = 1 for 0 < x < 1
s(x) = 1 - (1/y3))3
If events A and B are independent, what 3 things are true:
Pr(A∩B) = Pr(A)Pr(B)
Pr(A|B) = Pr(A)
Pr(B|A) = Pr(B)
if X = g(Y) and fx is PDF of X
then fY(y) = ?
fX(x(y)) |dx/dy|
∫0∞ yme-my dy = ?
Mean of exponential random variable with hazard rate m;
1/m
PDF of Poisson (Pr(N=n)) with lambda = L
fN(n) = (e-LLn)/n!
Pr(A∪B) = ?
Pr(A) + Pr(B) - Pr(A∩B)
X and Y are bivariate normally distributed with both means μx = μy = 0, σx2, σy2 and correlation pxy.
What is Var(Y|X=x)?
Var(Y|X=x) = (1-pxy2)(σy2)
Var(aX+bY+cZ) when given Cov between X Y and Z
a2Var(X) + b2Var(Y) + c2Var(Z) + 2abCov(X,Y) + 2acCov(X,Z) + 2bcCov(Y,Z)
Recall this key formula which relates Var(Y) to Var(Y|X)
Var(Y) = E(Var(Y|X))+Var(E(Y|X))
ChebyShevs Inequality?
Pr(|X-u|/σ > r) < 1/r2
X has CDF FX(x).
Let X(1), …, X(n) be ordered statistics.
What is FX(n)(x) and SX(n)(x)?
FX(n)(x) = (FX(x))n
SX(n)(x) = 1 - (FX(x))n
X has CDF FX(x).
Let X(1), …, X(n) be ordered statistics.
What is FX(1)(x) and SX(1)(x)?
FX(1)(x) = 1 - (1 - FX(x))n
SX(1)(x) = (1 - FX(x))n
… i think check NB
What does the hazard rate function equal in terms of the survival function s(x)?
d/dx -(ln(s(x)))
How to find E(X2|Y=y) when given fxy?
Integrate x2 fx(x|Y=y)
What is the PDF of an exponential distribution with hazard rate a?
ae-ax
Moment generating function of exponential function with hazard rate a?
MT(t) = a/(a-t)
convolution formula:
let Z = X + Y, what is fX+Y(z) in terms of fXY?
Integral of fXY(x,z-x) dx