Question

Suppose that X and Y have the following
joint probability density function.f (x, y) =
y, 0 < x < 5, y
> 0, x − 4 < y < x +
4 |

(a) |
Find E(XY). |

(b) |
Find the covariance between X and Y. |

Answer #1

Suppose that the joint probability density function of the
random variables X and Y is f(x, y) = 8 >< >: x + cy^2 0 ≤
x ≤ 1, 0 ≤ y ≤ 1 0 otherwise.
(a) Sketch the region of non-zero probability density and show
that c = 3/ 2 .
(b) Find P(X + Y < 1), P(X + Y = 1) and P(X + Y > 1).
(c) Compute the marginal density function of X and Y...

Suppose that X and Y have the following joint probability
density function. f (x, y) = 3 /146 *x, 0 < x < 5, y
> 0, x − 2 < y < x + 2 Find E(X).

Suppose that the random variables X and Y
have the following joint probability density function.
f (x, y) =
ce−5x − 7y, 0
< y < x.
(a)
Find the value of c.
(b)
Find P(X < 1/3 , Y <
2)

Suppose that the random variables X and Y
have the following joint probability density function.
f (x, y) =
ce−9x − 7y, 0
< y < x.
(a)
Find the value of c.
(b)
Find P(X <1/6
, Y < 1)

1. Let (X; Y ) be a continuous random vector with joint
probability density function
fX;Y (x, y) =
k(x + y^2) if 0 < x < 1 and 0 < y < 1
0 otherwise.
Find the following:
I: The expectation of XY , E(XY ).
J: The covariance of X and Y , Cov(X; Y ).

Suppose that X and Y have joint probability density function
given by: f(x, y) = 2 for 0 ≤ x ≤ 1 and 0 ≤ y ≤ x. What is Cov(X, Y
)?

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?

Suppose that the joint density function of X and
Y is given by
f (x, y) =
45 xe−3x(y +
5) x > 0,
y > 0.
(a)
Find the conditional density of X, given Y
= y.
(b)
Find the conditional density of Y, given X
= x.
(c)
Find P(Y > 5 | X = 4).

7. Suppose that random variables X and Y have a joint density
function given by: f(x, y) = ? + ? 0 ≤ ?≤ 1, 0 ≤ ? ≤ 1
(a) Find the density functions of X and Y, f(x) and f(y).
(b) Find E[X] and Var(Y).

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...

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