Question

X, Y, Z are zero mean correlated random variables with common correlation coefficient equal to - 1/2 and all variances equal to one.

a. Find the best linear estimate of Z in terms of X and Y ?

b. Find the best linear estimator for X in terms of Y and Z?

c. What are the minimum mean square estimation errors in the above cases?

Answer #1

2.33 X and Y are independent zero mean Gaussian random variables
with variances sigma^2 x, and sigma^2 y. Let Z = 1/2(X + Y) and W
=1/2 (X - Y) a. Find the joint pdf fz, w(z, w). b. Find the
marginal pdf f(z). c. Are Z and W independent?

3) Four statistically independent random variables, X, Y, Z, W
have means of 2, -1, 1, -2 respectively, variances of X and Z are 9
and 25 respectively, mean-square values of Y and W are 5 and 20
respectively. Define random variable V as: V = 2X - Y + 3Z - 2W,
find the mean-square value of V (with minimum math).

The random variables, X and Y , have the joint pmf
f(x,y)=c(x+2y), x=1,2 y=1,2 and zero otherwise.
1. Find the constant, c, such that f(x,y) is a valid pmf.
2. Find the marginal distributions for X and Y .
3. Find the marginal means for both random variables.
4. Find the marginal variances for both random variables.
5. Find the correlation of X and Y .
6. Are the two variables independent? Justify.

X and Y ar i.i.d. exponential random variables
with mean = 2. Let Z = X + Y. The
probability that Z is less than or equal to 3 is:

In Module 8, we discussed the correlation coefficient of two
random variables, say X and Y. Suppose X is the head circumference
of a person, and Y is the person's IQ score. If the correlation
coefficient is found to be 0.7943, please interpret this with a
brief statement. Use your own words!

The sample correlation coefficient is equal to the covariance of
x and y
divided by the square root of the product of
sx2 times sy2.
True
False

Consider three binary random variables X, Y, Z with domains {+x,
-x}, {+y, -y}, and {+z, -z}, respectively. Please use summation
notation
a) Express P(X) in terms of the joint distribution P(X,Y,Z)
b) Express P(X) in terms of P(Z), P(Y|Z), and P(X|Y, Z)
c) Expand the sums from part (b) to express the two elements of
P(X), (P(+x) and P(-x)) in terms of the individual probabilities
(e.g. P(-c) instead of P(C))

Let X be a random variable with a mean of 9 and a variance of
16. Let Y be a random variable with a mean of 10 and a variance of
25. Suppose the population correlation coefficient between random
variables X and Y is -0.4.
a) Find the mean of the random variable W = 3X - 5Y.
b) Find the standard deviation of the random variable Z = X +
Y

Let X and Y be two normal random variables. Given that the mean
of X is 6 and its standard deviation is equal to 1, and the mean of
Y is 2 with standard deviation 3. What is the probability that Z=
X-3Y is positive?.

15.1The probability density function of the X
and Y compound random variables is given below.
X
Y
1
2
3
1
234
225
84
2
180
453
161
3
39
192
157
Accordingly, after finding the possibilities for each value, the
expected value, variance and standard deviation; Interpret the
asymmetry measure (a3) when the 3rd moment (µ3 = 0.0005)
according to the arithmetic mean and the kurtosis measure
(a4) when the 4th moment (µ4 = 0.004) according to the
arithmetic...

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