EK381 T/F Exam2
Let X be a uniformly distributed random variable on [0,2], and let fY|X (y|x) be such that E[Y|X =x]=3x. Then, E[Y]=3
True
Let X is an exponential random variable with E[X] = 1. Then, the random variable Y = 3X is also an exponential random variable with Var[Y ] = 9
True
If W = 2X + 3 and Z = −Y + 2, then the correlation coefficients satisfy ρW,Z = −ρX,Z
True
If X and Y are jointly Gaussian, then E[X|Y = y] is a linear function of y plus a constant
True
fX|Y (x|y) = fY|X (y|x)
False
If ρX,Y =0,then E[XY]=E[X]E[Y]
True
Let X and Y be independent, each with mean 1 and variance 1. Then, E[X²Y²] = 4
True
Let X and Y be jointly Gaussian with Var[X]=Var[Y]. Define U =X+Y, V =X-Y. Then, U and V are independent.
True
Cov[X,X] = Var[X]
True
E[(X+Y)²]=E[X²]+2E[XY]+E[Y²]
True
E[XY]≤√E[X²]E[Y²]
True
For a<b<c, we have that P[a<X<c|X<b]=1- Fx(a)/Fx(b)
True
If Cov[X,Y]=0,then Cov[−X,Y]=0
True
Assume now that X is a Gaussian random variable with mean 0 and variance 1. Then, E[X⁴] = 0
False
If E[XY]=0, then ρX,Y =0
False
If P[X>Y]=1, then fX(a)>fY(a) for every choice of a.
False
If X and Y are independent, then P{X>a}∩{Y >a}=1−FX(a)FY(a)
False
If X and Y are independent, then Var[XY ] =E[X²]E[Y²] − (E[X²])²(E[Y²])²
False
If X and Y are independent, then Var[aX + bY ] = aVar[X] + bVar[Y]
False
If Cov[X,Y]=E[XY],then either E[X]=0 or E[Y]=0 or both are zero
True
If X, Y are jointly continuous random variables, and Z = 5 Y + X , then Var[Z] = Var[X]+2Cov[X,Y]+Var[Y]
False
If b > a, then FX(b) > FX(a)
False
If fX(a) ≥ fX(b), then FX(a) ≥ FX(b)
False
If E[X]=E[Y]=0,then Var[X+Y]=E[X²]+E[Y²]+2E[XY]
True
If ρX,Y = ¼, then Y = X/4 +b from some constant b
False
If ρX,Y > 0, then X and Y have the same sign with probability at least ½
False
P[A∩B]≤P[A]P[B]
False
P[X²>a]= ∫fx(x)dx √a to ∞
False
If E[X|Y =y]=0,then E[X]=0
True
If E[X|Y =y]=y²,then E[X]=Var[Y]+(E[Y])²
True
If FX(a) > FX(b), then a > b
True
If Var[X]=a², and Var[Y]=b², then Cov[X,Y]≤ab
True
