Question

Suppose you choose two numbers x and y, independently at random from the interval [0, 1]. Given that their sum lies in the interval [0, 1], find the probability that (a) |x − y| < 1. (b) xy < 1/2. (c) max{x, y} < 1/2. (d) x 2 + y 2 < 1/4. (e) x > y

Answer #1

Choose two numbers X and Y independently at
random from the unit interval [0,1] with the uniform density. The
probability that
X^2+Y^2>0.49
THE ANSWER IS NOT .0192129

Choose real numbers X and Y uniformly and independently in [0;
1]. What is the
probability that the quadratic equation a2 + Xa + Y = 0 has two
distinct real
solutions a1 and a2?
Hint: Draw a picture in the XY -plane.

Choose two numbers ? and ? independently at random from the unit
interval [0,1] with the uniform density. The probability that
4⋅?+8⋅? < 0.8 is

1. Consider the following optimization problem. Find two
positive numbers x and y whose sum is 50 and whose product is
maximal. Which of the following is the objective function?
A. xy=50
B. f(x,y)=xy
C. x+y=50
D. f(x,y)=x+y
2. Consider the same optimization problem. Find two positive
numbers x and y whose sum is 50 and whose product is maximal. Which
of the following is the constraint equation?
A. xy=50
B. f(x,y)=xy
C. x+y=50
D. f(x,y)=x+y
3. Consider the same...

Twenty numbers are uniformly and independently selected from the
interval (0, 1). Use the Central Limit Theorem to find the
approximate probability that their sum is at least 8. Express your
answer in terms of F, the standard normal distribution
function.
A. F(1.78)
B. 1 − F(−1.38)
C. F(2.16)
D. 1 − F(−1.55)

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...

f_X,Y(x,y)=xy 0<=x<=1, 0<=y<=2
f_X(x)=2x 0<=x<=1, f_Y(y)=y/2
0<=y<=2 choose the all correct things.
a. E[X]=1/2
b. E[XY]=8/9
c. COV[X,Y]=1
d. correlation coefficiet =1

Suppose you choose a real number X from the interval
[3,16] with the density function
f(x)=Cx,
where C is a constant.
a) Find C. Remember that if you integrate a density
function over the entire sample space interval, you should get
1.
b) Find P(E), where
E=[a,b] is a subinterval of [3,16]
(as a function of a and b ).
c) Find P(X>4)
d) Find P(X<14)
e) Find P(X^2−18X+56≥0)
Note: You can earn partial credit on this
problem.

(a) Given two independent uniform random variables X, Y in the
interval (−1, 1), find E |X − Y |.
(b) Let X, Y be as in (a). Find the support and density of the
random variable Z = |X − Y |.
(c) From (b), compute the mean of Z and check whether you get
the same answer as in (a)

LetX,Ybe a pair of continuous random variables with joint
density functionf(x,y) ={kxy,for 0≤x≤1 and 0≤y≤1,0otheriwse.
(a) Findk. (4 pts.)
(b) Find the marginal distribution ofX,fX(x). (4 pts.)
(c) Find P(X >0.5). (4 pts.)
(d) Find E(XY). (4 pts.)
(e) Find Cov(X,Y). (4 pts.)

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