Question

Let X,Y be independent [0,1]-uniform. Calculate expected values
of Z_{1}=XY/(X+1), Z_{2}= X/(Y+1),
Z_{3}=(x+y)(x-2y). Calculate r[(X+Y), (X-Y)].

Answer #1

For the bar, R, and S table provided below, calculate
the Z1, Z2, Z3, W1, W2, W3, and central line of W and Z bar chart
and control limits. Target grand average (X-double bar) is 7.0 and
target R-bar is 0.25.
sub1 sub2 sub3
Xbar 7.020 7.080 7.000
R 0.300 0.500 0.300
S 0.130 0.205 0.1224

Let X and Y be independent and identical uniform distribution on
[0,1]. Let Z=min(X, Y). Find E[Y-Z]. What is the probability
Y=Z?

Suppose that X and Y are independent Uniform(0,1) random
variables. And let U = X + Y and V = Y .
(a) Find the joint PDF of U and V
(b) Find the marginal PDF of U.

Let x = [1, 1]T , y = [1, 1]T ∈ R 2 and let f : R 2 =⇒ R 2 with
f(z) =z1.x + z2.y for any z = [z1, z2] T ∈ R 2 . Further, z = g(r)
= [r 2 , r3 ] where r ∈ R . Show how chain rule is applied here
giving major steps of the calculation, write down the expression
for ∂f ∂r , and also evaluate ∂f/ ∂r at...

A random variable X has probability density function f(x)
defined by f(x) = cx−6 if x > 1, and f(x) = 0, otherwise.
a. Find the constant c.
b. Calculate E(X) and Var(X).
c. Now assume Z1, Z2, Z3, Z4 are independent RVs whose
distribution is identical to that of X. Compute E[(Z1 +Z2 +Z3
+Z4)/4] and Var[(Z1 +Z2 +Z3 +Z4)/4].
d. Let Y = 1/X, using the formula to find the pdf of Y.

Let X be continuous uniform (0,1) and Y be exponential (1). Let
O1 = min(X,Y) and O2 = max(X,Y) be the order statistics of ,Y. Find
the joint density of O1, O2.

Uncorrelated and Gaussian does not imply independent unless
jointly Gaussian. Let X ∼N(0,1) and Y = WX, where p(W = −1) = p(W =
1) = 0 .5. It is clear that X and Y are not independent, since Y is
a function of X. a. Show Y ∼N(0,1). b. Show cov[X,Y ]=0. Thus X and
Y are uncorrelated but dependent, even though they are Gaussian.
Hint: use the deﬁnition of covariance cov[X,Y]=E [XY] −E [X] E [Y ]
and...

X and Y are independent variables, with X having a uniform (0,1)
distribution and Y being an exponential random variable with a mean
of 1.
Given this information, find P(max{X,Y} > 1/2)

X and Y are independent and identically distributed variables
uniform over [0,1]. Find PDF of A=Y/X

1.
a) Let: z=rcis(t). Enter an argument of -z.
b) Let:z1=4cis(3π/4) and z2=2cis(π/3).
Calculate z1*(/z2) in polar form. Find its modulus and
principle argument. Calculate (z1/z2) in polar form. Find its
modulus and principle argument.
where /z2: the conjugate of z2

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