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5. Let (X, Y) be a uniformly distributed random point on the quadrilateral D with vertices...
Exercise 6.34. Let (X,Y) be a uniformly distributed random point on the quadri- lateral D with...
Exercise 6.34. Let (X,Y) be a uniformly distributed random point on the quadri- lateral D with vertices (0,0), (2,0), (1,1) and (0,1). (a) Find the joint density function of (X,Y) and the marginal density functions of X and Y. (b) Find E[X] and E[Y]. (c) Are X and Y independent?
3. The pair of random variables X and Y is uniformly distributed on the interior of the triangle with the vertices whose coordinates are (0,0), (0,2), and (2,0) (i.e., the joint density is equal to a...
3. The pair of random variables X and Y is uniformly distributed on the interior of the triangle with the vertices whose coordinates are (0,0), (0,2), and (2,0) (i.e., the joint density is equal to a constant inside the triangle and zero outside). (a) (10 points) Find P(Y+X< 1). (b) (10 points) Find P(X = Y). (c) (10 points) Find P(Y > 1X = 1/2).
3. The pair of random variables X and Y is uniformly distributed on the interior...
(3) A pair of random variables (X, Y) is distributed uniformly on the triangle with vertices...
(3) A pair of random variables (X, Y) is distributed uniformly on the triangle with vertices (0,0), (2,0) and (0. Find EX, EY, Cov(X,Y), E(max{X,Y)), P(X> Y), P(X 2 Y)
2. A random vector (X,Y) is uniformly distributed in a triangle with vertices ABC, having coordinates...
2. A random vector (X,Y) is uniformly distributed in a triangle with vertices ABC, having coordinates A(0,0), B(2.0), C(0,1). The joint density f(x,y) is given by the formula 141) - f(x,y) = { s 15 inside the triangle outside
3. The pair of random variables X and Y is uniformly distributed on the interior of...
3. The pair of random variables X and Y is uniformly distributed on the interior of the triangle with the vertices whose coordinates are (0,0), (0,2), and (2,0) (i.e., the joint density is equal to a constant inside the triangle and zero outside). (a) (10 points) Find P(Y+X< 1). (b) (10 points) Find P(X = Y). (c) (10 points) Find P(Y > 1X = 1/2).
Exercise 10.33. Let (X,Y) be uniformly distributed on the triangleD with vertices (1,0), (2,0) and (0,1), as in Example...
Exercise 10.33. Let (X,Y) be uniformly distributed on the
triangleD with vertices (1,0), (2,0) and (0,1), as in Example
10.19. (a) Find the conditional probability P(X ≤ 1 2|Y =y). You
might first deduce the answer from Figure 10.2 and then check your
intuition with calculation. (b) Verify the averaging identity for
P(X ≤ 1 2). That is, check that P(X ≤ 1 2)=:∞ −∞ P(X ≤ 1 2|Y
=y)fY(y)dy.
Example 10.19. Let (X, Y) be uniformly distributed on the...
5. Let X be uniformly distributed over (0,1). a) Find the density function of Y =...
5. Let X be uniformly distributed over (0,1). a) Find the density function of Y = ex. b) Let W = 9(X). Can you find a function g for which W is an exponential random variable? Explain.
9 0/1 point (graded) Let X be distributed uniformly over 1,1, andlet y- Xw.p. 3/4, l-X...
9 0/1 point (graded) Let X be distributed uniformly over 1,1, andlet y- Xw.p. 3/4, l-X w.p. 1/4, where w.p. means "with probability". Find Cov (X,Y) 2 Submit You have used 1 of 4 attempts
3. Consider two random variables X and Y, whose joint density function is given as follows....
3. Consider two random variables X and Y, whose joint density function is given as follows. Let T be the triangle with vertices (0,0), (2,0), and (0,1). Then if (x, y for some constant K (a) (2 pts.) Find the constant K (b) (4 pts.) Find P(X +Y< 1) and P(X > Y). (c) (4 pts.) Find the marginal densities fx and fy. Conclude that X and Y are not independent
15. Let X and Y denote the lengths of life, in hundreds of hours, for co...
15. Let X and Y denote the lengths of life, in hundreds of hours, for co ponents of typesI and types II, respectively in an electronic system. The joint density of X and Y is given by Bre" (z +v)/2 f(z, y) = otherwise Find the probability that a component of type II will have a life lenght in excess of 200 hours. 16. Let the random variables X and Y have the joint p.d.f a. f(z,y)=ī, for (z,y) =...
Exercise 6.34. Let (X,Y) be a uniformly distributed random point on the quadri- lateral D with vertices (0,0), (2,0), (1,1) and (0,1). (a) Find the joint density function of (X,Y) and the marginal density functions of X and Y. (b) Find E[X] and E[Y]. (c) Are X and Y independent?
3. The pair of random variables X and Y is uniformly distributed on the interior of the triangle with the vertices whose coordinates are (0,0), (0,2), and (2,0) (i.e., the joint density is equal to a constant inside the triangle and zero outside). (a) (10 points) Find P(Y+X< 1). (b) (10 points) Find P(X = Y). (c) (10 points) Find P(Y > 1X = 1/2).
3. The pair of random variables X and Y is uniformly distributed on the interior...
(3) A pair of random variables (X, Y) is distributed uniformly on the triangle with vertices (0,0), (2,0) and (0. Find EX, EY, Cov(X,Y), E(max{X,Y)), P(X> Y), P(X 2 Y)
2. A random vector (X,Y) is uniformly distributed in a triangle with vertices ABC, having coordinates A(0,0), B(2.0), C(0,1). The joint density f(x,y) is given by the formula 141) - f(x,y) = { s 15 inside the triangle outside
3. The pair of random variables X and Y is uniformly distributed on the interior of the triangle with the vertices whose coordinates are (0,0), (0,2), and (2,0) (i.e., the joint density is equal to a constant inside the triangle and zero outside). (a) (10 points) Find P(Y+X< 1). (b) (10 points) Find P(X = Y). (c) (10 points) Find P(Y > 1X = 1/2).
Exercise 10.33. Let (X,Y) be uniformly distributed on the
triangleD with vertices (1,0), (2,0) and (0,1), as in Example
10.19. (a) Find the conditional probability P(X ≤ 1 2|Y =y). You
might first deduce the answer from Figure 10.2 and then check your
intuition with calculation. (b) Verify the averaging identity for
P(X ≤ 1 2). That is, check that P(X ≤ 1 2)=:∞ −∞ P(X ≤ 1 2|Y
=y)fY(y)dy.
Example 10.19. Let (X, Y) be uniformly distributed on the...
5. Let X be uniformly distributed over (0,1). a) Find the density function of Y = ex. b) Let W = 9(X). Can you find a function g for which W is an exponential random variable? Explain.
9 0/1 point (graded) Let X be distributed uniformly over 1,1, andlet y- Xw.p. 3/4, l-X w.p. 1/4, where w.p. means "with probability". Find Cov (X,Y) 2 Submit You have used 1 of 4 attempts
3. Consider two random variables X and Y, whose joint density function is given as follows. Let T be the triangle with vertices (0,0), (2,0), and (0,1). Then if (x, y for some constant K (a) (2 pts.) Find the constant K (b) (4 pts.) Find P(X +Y< 1) and P(X > Y). (c) (4 pts.) Find the marginal densities fx and fy. Conclude that X and Y are not independent
15. Let X and Y denote the lengths of life, in hundreds of hours, for co ponents of typesI and types II, respectively in an electronic system. The joint density of X and Y is given by Bre" (z +v)/2 f(z, y) = otherwise Find the probability that a component of type II will have a life lenght in excess of 200 hours. 16. Let the random variables X and Y have the joint p.d.f a. f(z,y)=ī, for (z,y) =...
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