Question

Let X and Y have joint pdf f(x,y)=k(x+y), for 0<=x<=1 and 0<=y<=1.

a) Find k.

b) Find the joint cumulative density function of (X,Y)

c) Find the marginal pdf of X and Y.

d) Find Pr[Y<X^{2}] and Pr[X+Y>0.5]

Answer #1

STAT 190 Let X and Y have the joint probability density function
(PDF), f X,Y (x, y) = kx, 0 < x < 1, 0 < y < 1 -
x^2,
= 0, elsewhere,
where k is a constant.
1) What is the value of k.
2)What is the marginal PDF of X.
3) What is the E(X^2 Y).

1. Let (X,Y ) be a pair of random variables with joint pdf given
by f(x,y) = 1(0 < x < 1,0 < y < 1).
(a) Find P(X + Y ≤ 1).
(b) Find P(|X −Y|≤ 1/2).
(c) Find the joint cdf F(x,y) of (X,Y ) for all (x,y) ∈R×R.
(d) Find the marginal pdf fX of X. (e) Find the marginal pdf fY
of Y .
(f) Find the conditional pdf f(x|y) of X|Y = y for 0...

1. for 0<= x <=3 0<=x<=1 f(x,y) = k(x^2y+ xy^2)
a. Find K joint probablity density function.
b. Find marginal distribution respect to x
c. Find the marginal distribution respect to y
d. compute E(x) and E(y) e. compute E(xy)
f. Find the covariance and interpret the result.

4. Let X and Y be random variables having joint probability
density function (pdf) f(x, y) = 4/7 (xy − y), 4 < x < 5 and
0 < y < 1
(a) Find the marginal density fY (y).
(b) Show that the marginal density, fY (y), integrates to 1
(i.e., it is a density.)
(c) Find fX|Y (x|y), the conditional density of X given Y =
y.
(d) Show that fX|Y (x|y) is actually a pdf (i.e., it integrates...

Let X and Y have the joint probability density function f(x, y)
= ⎧⎪⎪ ⎨ ⎪⎪⎩ ke−y , if 0 ≤ x ≤ y < ∞, 0, otherwise. (a) (6pts)
Find k so that f(x, y) is a valid joint p.d.f. (b) (6pts) Find the
marginal p.d.f. fX(x) and fY (y). Are X and Y independent?

The joint probability density function (pdf) of X and Y is given
by
f(x, y) = cx^2 (1 − y), 0 < x ≤ 1, 0 < y ≤ 1, x + y ≤
1.
(a) Find the constant c.
(b) Calculate P(X ≤ 0.5).
(c) Calculate P(X ≤ Y)

Let X and Y have the joint pdf f(x,y) = 6*(x^2)*y for 0 <= x
<= y and x + y <= 2.
What is the marginal pdf of X and Y?
What is P(Y < 1.1 | X = 0.6)?
Are X and Y dependent random variables?

Let X and Y have the joint PDF (i really just need g and
h if that makes it easier)
f(x) = { c(y + x^2) 0 < x < 1 and 0 < y < 1 ; 0
elsewhere
a) Find c such that this is a PDF.
b) What is P(X ≤ .4, Y ≤ .2) ?
C) Find the Marginal Distribution of X, f(x)
D) Find the Marginal Distribution of Y, f(y)
E) Are X and Y...

Let f(x, y) = c/x, 0 < y < x < 1 be the joint density
function of X and Y .
a) What is the value of c?
a) 1 b) 2 c) 1/2 d) 2/3 e) 3/2
b)what is the marginal probability density function of X?
a) x/2 b)1 c)1/x d)x e)2x
c)what is the marginal probability density function of Y ?
a) ln y b)−ln y c)1 d)y e)y2
d)what is E[X]?
a)1 b)2 c)4 d)1/2 e)1/4

The joint probability density function (pdf) describing
proportions X and Y of two components in a chemical blend are given
by f(x, y) = 2, 0 < y < x ≤ 1.
(a) Find the marginal pdfs of X and Y.
(b) Find the probability that the combined proportion of these
two components is less than 0.5.
(c) Find the conditional probability density function of Y given
X = x. (d) Find E(Y | X = 0.8).

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