. Let X and Y be random variables. The conditional
variance of Y given X, denoted Var(Y | X),
is defined as
Var(Y | X) = E[Y
2
| X] − E[Y | X]
2
.
Show that Var(Y ) = E[Var(Y | X)] + Var(E[Y | X]). (This equality
you are showing is known
as the Law of Total Variance). Hint: From the Law of Total
Expectation, you get Var(Y ) =
E[Y
2
] − E[Y ]
2 = E
h
E[Y
2
| X]
i
− E
h
E[Y | X]
i2
.
. Let X and Y be random variables. The conditional variance of Y given X, denoted...
For random variables X and Y with finite variance, the law of total variance states that Var(X) E(Var(X|Y)) + Var(E(XTY)) variance analogue of E Write out a formula that For each n, let Varn be the conditional relates two such conditional variances as the formula for the iteration condition relates two conditional expectations
O RANDOM VARIABLES AND DISTRIBUTIONS Expectation and variance of a random variable Let X be a random variable with the following probability distribution: Value x of X P(X-) 0.35 0.40 0.10 0.15 10 0 10 20 Find the expectation E (X) and variance Var(X) of X. (If necessary, consult a list of formulas.) Var(x) -
The conditional variance of X, given Y, is defined by Prove the conditional variance formula, namely, Var(X) E[Var(X|Y)] Var(E[XYl) Use this to obtain Var(X) in Example 1 S(B) and check your result by differentiating the generating function
The random variables X and Y have the joint PDF fx,y(x,y)=0.5, if x>0 and y>0 and xtys2, and 0 otherwise. Let A be the event Ys1) and let B be the event (Y>X). (You can use rational numbers like 3/5 for your answers.) 1. Calculate P(BIA). 2. Calculate fxıy(xlO.9) fxIY(0.39820710.9) 3. Calculate the conditional expectation of X, given that Y=1.8 4, Calculate the conditional variance of X, given that Y=1.4 5. Calculate fxlB(x) fXIB(0.11) 6. Calculate E[XY]. 7. Calculate the...
MA2500/18 Section B (Answer THREE questions) 6. Let X and Y be jointly continuous random variables defined on the same prob- ability space, let fx.y denote their joint PDF, and let fx and fy respectively denote their marginal PDFs (a) Let z be a fixed value such that fx(x) >0. Write down expressions for 12] (i) the conditional PDF of Y given X = z, and (i) the conditional expectation of Y given X (b) State and prove the law...
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...
(3) 18 pts] Let Ya and Y, denote Bernoulli random variables from two different populations, denoted a and b. Suppose that E(%)-Pa and E(%)-pb. A random sample of size na is chosen from population a, with sample average denoted pa, and a random sample of size nb is chosen from population b, with sample average denoted Suppose the sample from population a is independent of the sample from population b. (a) Show that E(Pi) P and var(P) p pi)/n, for...
Given random variables X1, X2, Y with E[Y | X1, X2] = 5X1 + X1X2 and E[Y 2 | X1, X2] = 25X2 1X2 2 + 15, find E[(X1Y + X2) 2 | X1, X2]. ㄨ竺Bin(2.1/4). Suppose X and Y are independent random variables. Find the expected value of YX. Hnt: Consider conditioning on the events (X-j)oj0,1,2. 9. Given random variables XI,X2, Y with E'Y | XiN;|-5X1 + X1X2 and Ep2 1 X1,X2] 25XX15, find 10. Let X and Y...
1. Let X and Y b e random variables, with μΧ = E(X), μΥ = E(Y), σ炙= Var(X) and σ Var(Y) (2) Let Ỹ be a linear function of X, ie. Ỹ = +51X where bo and bl are fixed real numbers. We want to minimize the discrepancy of Y from Y, i.e. minimizing the quantity (a) Find the values of bo and bi that minimizes Q (b) Use (a) to show that the minimal value of Q is σ....
(3) 18 pts] Let Ya and Y, denote Bernoulli random variables from two different populations, denoted a and b. Suppose that E(%)-Pa and E(%)-pb. A random sample of size na is chosen from population a, with sample average denoted pa, and a random sample of size nb is chosen from population b, with sample average denoted Suppose the sample from population a is independent of the sample from population b (a) Show that E(Pi) Pi and var(Pi)-P( pi)/n, for j...