a. Show = b. Show E{(X-EX)(Y-EY)}=E{(X-EX)Y}
Let X and Y be two independent random variables. Show that Cov (X, XY) = E(Y) Var(X).
(e) (2 pts) Show that, if X and Y are two uncorrelated (ie. EXY = EXEY) Bernoulli (Indicator) random variables then they are independent.
which of the following is a potential function for F(x,y,z)= < y2 +y?ex?,x2 + 2ye*?,xy + xy?e *V> f(x,y,z) = xyz + y2exyz f(x,y,z) = xyz + y2e*+2 b. F(x,y,z) has a potential function but it is not one of the other choices. F(x,y,z) does not have a potential function. d. f(x,y,z) = xyz + y2exZ e.
Let xi be independent. E(xi)=0. Var(xi)= sigma ^2
Cov(x,y) = E(XY) - ExEy
Use this fact and apply it to this example ! Do not use
anything that has not been giving. I’m having difficulties
completing this problem. Check pictures to see how I done a
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Problem 1: Show E[Y-G(X)|2] is minimized using G(X) = EX ing G(X) EYAX
Problem 1: Show E[Y-G(X)|2] is minimized using G(X) = EX ing G(X) EYAX
X and Y are random variables (a) Show that E(X)=E(B(X|Y)). (b) If P((X x, Y ) P((X x})P({Y y)) then show that E(XY) = E(X)E(Y), i.e. if two random variables are independent, then show that they are uncorrelated. Is the reverse true? Prove or disprove (c) The moment generating function of a random variable Z is defined as ΨΖφ : Eez) Now if X and Y are independent random variables then show that Also, if ΨΧ(t)-(λ- (d) Show the conditional...
there is no need to introduce new variables
2. Consider a joint pdf Find: (a) E(X|Y = y). (b) E(Y|X =x). (c) E(XY)
2. Consider a joint pdf Find: (a) E(X|Y = y). (b) E(Y|X =x). (c) E(XY)
C2.3 Let X and Y be random variables with finite variance, so that EX2o0 (i) Show that E(X) - (EX) E(X - EX)2, and hence that the variance of (ii) By considering (|XI Y)2, or otherwise, show that XY has finite expecta- (iii) Let q(t) = E(X + tY)2. Show that q(t)2 0, and by considering the roots of and EY2 < oo. X is always non-negative. tion the equation q(t) 0, deduce that
For boolean algebra can you use an identity but make everything
opposite? NOTs
Ex. x+!xy = x+y Can you do !x+x!y = !x+!y ???
or Ex. xy+!xy =y (minimization) Can you do !x!y + x!y = !y
???
Note this is boolean algebra. ! means NOT. Imagine a line above
the particular letter to the right of it.
<- like this, x bar.