Suppose that the joint probability density function of ˜ (X, Y) is given by:´
f X,Y (x,y) = 4x/y3 I(0.1)(x), I (1, ∞)(y).
Calculate
a) P(1/2 < X < 3/4, 0 < Y ≤ 1/3).
b) P(Y > 5).
c) P(Y > X).
Suppose X and Y have joint probability density function f(x,y) =
6(x-y) when 0<y<x<1 and f(x,y) = 0 otherwise.
(a) Indicate with a sketch the sample space in the x-y plane
(b) Find the marginal density of X, fX(x)
(c) Show that fX(x) is properly normalized, i.e., that it
integrates to 1 on the sample space of X
(d) Find the marginal density of Y, fY(y)
(e) Show that fY(y) is properly normalized, i.e., that it
integrates to 1 on...
Suppose that we have two random variables (Y,X) with joint
probability density function f(y,x). Consider the following
estimator of the intercept of the Best Linear Predictor:
A = ?̅ - B • ?̅ ,
where ?̅ is the sample mean of y, ?̅ is the sample mean of x, and B
is the sample covariance of Y and X divided by the sample variance
of X.
Identify the probability limit of A (if any). For each step in
your derivation,...
. The joint probability density function of X and Y is given
by
?(?, ?) = { ??^2? ?? 0 ≤ ? ≤ 2, 0 ≤ ?, ??? ? + ? ≤ 1
0 ??ℎ??????
(a) Determine the value of c.
(b) Find the marginal probability density function of X and
Y.
(c) Compute ???(?, ?).
(d) Compute ???(?^2 + ?).
(e) Determine if X and Y are independent
Consider a continuous random vector (Y, X) with joint
probability density function
f(x, y) = 1
for 0 < x < 1, x < y < x + 1.
What is the marginal density of X and Y? Use this to compute
Var(X) and Var(Y)
Compute the expectation E[XY]
Use the previous results to compute the correlation Corr (Y,
X)
Compute the third moment of Y, i.e., E[Y3]
Consider a continuous random vector (Y, X) with joint
probability density function f(x, y) = 1 for 0 < x < 1, x
< y < x + 1.
A. What is the marginal density of X and Y ? Use this to compute
Var(X) and Var(Y).
B. Compute the expectation E[XY]
C. Use the previous results to compute the correlation Corr(Y,
X).
D. Compute the third moment of Y , i.e., E[Y3].
5. Suppose that X and Y have the following joint probability
distribution:
f(x,y)
x
2
4
y
1
0.10
0.15
2
0.20
0.30
3
0.10
0.15
Find the marginal distribution of X and Y.
Find the expected value of g(x,y) = xy2 or find E(xy2).
Find (x and (y.
Find Cov(x,y)
Find the correlations ρ(x,y)
3.
The length of life X, in days, of a heavily used electric motor
has probability density function
Find the probability that the motor has...
Question 3
Suppose that X and Y have the following joint probability
distribution:
f(x,y)
x
0
1
2
y
0
0.12
0.08
0.06
1
0.04
0.19
0.12
2
0.04
0.05
0.3
Find the followings:
E(Y)=
Var(X)=
Cov(X,Y)=
Correlation(X,Y)=
If the joint probability density function of the random
variables X and Y is given by f(x, y) = (1/4)(x + 2y) for 0 < x
< 2, 0 < y < 1, 0 elsewhere
(a) Find the conditional density of Y given X = x, and use it to
evaluate P (X + Y/2 ≥ 1 | X = 1/2)
(b) Find the conditional mean and the conditional variance of Y
given X = 1/2
(c) Find the variance...
Let X and Y have the following joint density function
f(x,y)=k(1-y) , 0≤x≤y≤1.
Find the value of k that makes this a probability density
function.
Compute the probability that P(X≤3/4, Y≥1/2).
Find E(X).
Find E(X|Y=y).