Table y x 0 1 2 3 4 0 008 007 006 001 001 1 006 010

table:
Y
X
0 1 2
0 0.08 0.07 0.06 0.01
1 0.06 0.10 0.12 0.05
2 0.05 0.06 0.09 0.04
3 0.02 0.03 0.03 0.03
Determine each of the following probabilities:
(a) 𝑃(? = 2) (b) 𝑃(? ≥ 2) (c) 𝑃(? ≤ 2 and ? ≤ 2)
(d) 𝑃(? = ?) (e) 𝑃(? > ?)
176 . [DeGroot, P129, #3] Suppose that X and Y have a discrete joint distribution for which the joint p.f. is
defined as follows:
?(?, ?) = {
?|? + ?| if ? = −2, −1, 0, 1, 2 and ? = −2, −1, 0, 1, 2
0 otherwise
Determine
(a) the value of the constant c ; (b) 𝑃(? = 0 and ? = −2)
(c) 𝑃(? = 1) ; (d) 𝑃(|? − ?| ≤ 1)
177 . [DeGroot, P129, #1] Suppose that the joint p.d.f. of a pair of random variables
(?, ?)
is constant on
the rectangle where 0 ≤ ? ≤ 2 and 0 ≤ ? ≤ 1 , and suppose that the p.d.f. is 0 off of this rectangle.
(a) Find the constant value of the p.d.f. on the rectangle.
(b) Find 𝑃(? ≥ ?)
178 . [DeGroot, P129, #4] Suppose that X and Y have a continuous joint distribution for which the joint
p.d.f. is defined as follows:
?(?, ?) = {
??
2
if 0 ≤ ? ≤ 2 and 0 ≤ ? ≤ 1
0 otherwise
Determine
(a) the value of the constant c ; (b) 𝑃(? + ? > 2)
(c) 𝑃(? < 1 2 ⁄ ) ; (d) 𝑃(? ≤ 1)
(e) 𝑃(? = 3?)
179 . [DeGroot, P129, #5] Suppose that the joint p.d.f. of two random variables X and Y is as follows:
?(?, ?) = {
?(?
2
+ ?) if 0 ≤ ? ≤ 1 − ?
0 otherwise
Determine
(a) the value of the constant c ; (b) 𝑃(0 ≤ ? ≤ 1 2
⁄ )
(c) 𝑃(? ≤ ? + 1) ; (d) 𝑃(? = ?
2
180 . [DeGroot, P129, #6] Suppose that a point
(?, ?)
is chosen at random from the region S in the xy
plane containing all points
(?, ?)
such that ? ≥ 0 , ? ≥ 0 , and 4? + ? ≤ 4 . Determine the joint p.d.f.
of X and Y .
3 4
0.01
0.02
0.03
0.04
;
.
.
;
;
.
2
;
)
.
-

20
181 . [DeGroot, P129, #7] Suppose that a point
(?, ?)
is to be chosen from the square S in the xy -plane
containing all points (?, ?) such that 0 ≤ ? ≤ 1 and 0 ≤ ? ≤ 1 . Suppose that the probability that the
chosen point will be the corner
(0, 0)
is 0.1, the probability that it will be the corner
(1, 0)
is 0.2, the
probability that it will be the corner
(0, 1)
is 0.4, and the probability that it will be the corner
(1, 1)
0.1. Suppose also that if the chosen point is not one of the four corners of the square, then it will be an
interior point of the square and will be chosen according to a constant p.d.f. over the interior of the
square. Determine
(a) 𝑃(? ≤ 1 4 ⁄ ) (b) 𝑃(? + ? ≤ 1)
182 . [Strait, P251, #10] Suppose that X and Y are continuous random variables with joint probability
density function given by
?(?, ?) = {
(9 4 ⁄ )? 2
?
2
if − 1 < ? < 1 and − 1 ≤ ? ≤ 1
0 elsewhere
Find:
(a) 𝑃(? + ? ≥ 1) (b) 𝑃(? ≥ 0.5)
183 . [Strait, P252, #15] Let X and Y be discrete random variable with joint probability function f defined
by the following values: ?(−1, 0) = .2 , ?(−1, 1) = .1 , ?(−1, 2) = .05 , ?(0, 0) = .05 , ?(0, 1) = .2
?(0, 2) = .3 , ?(3, 0) = .02 , ?(3, 1) = .04 , and ?(3, 2) = .04
(a) Find the marginal probability function of X.
(b) Find the marginal probability function of Y.
(c) Find 𝑃(−1 ≤ ? ≤ 2) , 𝑃(−1 ≤ ? ≤ 2)
184 . [DeGroot, P140, #2] Suppose that X and Y have a discrete joint distribution for which the joint p.f. is
defined as follows:
?(?, ?) = {
1
30
(? + ?)
if ? = 0, 1, 2 and ? = 0, 1, 2, 3
0 otherwise
(a) Determine the marginal p.f.’s of X and Y.
(b) Are X and Y independent?
185 . [DeGroot, P140, #3] Suppose that X and Y have a continuous joint distribution for which the joint
p.d.f. is defined as follows:
?(?, ?) = {
3
2
?
2
if 0 ≤ ? ≤ 2 and 0 ≤ ? ≤ 1
0 otherwise
(a) Determine the marginal p.d.f.’s of X and Y.
(b) Are X and Y independent?
is
,
.
.