Let A be the triangle in the two-dimensional plane with vertices (0, 0), (0, 1), and...
Let A be the triangle in the two-dimensional plane with vertices (0, 0), (0, 1), and (1, 0). Let (X, Y ) be chosen uniformly from this area, that is, (X, Y ) ∼ Unif(A).
(a) What is the probability that X ≤ 1/3?
(b) What is the probability that Y ≥ 1/2?
(c) Conditioned on X ≤ 1/3, what is the probability that Y ≥ 1/2?
(d) Are the events X ≤ 1/3 and Y ≥ 1/2 independent?
Homework Answers
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Let A be the triangle in the two-dimensional plane with vertices
(0, 0), (0, 1), and...
Let C be a triangle in the x-y plane with vertices (x1,y1), (x2y2) and (x3,y3) arranged so that C...
Let C be a triangle in the x-y
plane with vertices (x1,y1), (x2y2) and (x3,y3) arranged so that C
is positively-oriented.
Let C be a triangle in the xy-plane with vertices (x,y), (z2,p), and (z3,U3) arranged so that C is positively-oriented. a.) Sketch such a triangle and indicate its orientation. b.) Apply Green's Theorem to compute the area of the triangle as a (sum of) path integral(s) around the boundary. Get a formula for area in terms of the coordinates...
Let C be a triangle in the ry-plane with vertices (ıv). (2.92), and (T3, Vs) arranged so that C' ...
Let C be a triangle in the ry-plane with vertices (ıv). (2.92), and (T3, Vs) arranged so that C' is positively-oriented a.) Sketch such a triangle and indicate its orientation. b.) Apply Green's Theorem to compute the area of the triangle as a (sum of) path integral(s) around the boundary. Get a formula for area in terms of the coordinates zi and y. This formula is sometimes referred to as the Shoelace formula for area. c.) Use the formula you...
(1 point) Let T'be the region inside the triangle with vertices (0, 0), (3, 0) and...
(1 point) Let T'be the region inside the triangle with vertices (0, 0), (3, 0) and (3, 2), and let f(x, y) be the function which is 0 outside of T and f(x, y)-38 SK y Ior (x, y) inside T. 3888 Then E(XY)- 4.8
Suppose the two-dimensional random variable (X, Y ) is uniformly distributed over the triangle of the figure.
Suppose the two-dimensional random variable (X, Y ) is uniformly distributed over the triangle of the figure.a) What is f.d.p.c. of (X,Y). Calculate P(0 < X ≤ 1, Y > 1). Make a graphic sketch of the regionthat you used to calculate the probability. b) Determine the marginal distributions. (X, Y ) are independent?c) Find E[X] ,V AR[X], E[Y ] e V AR[Y ];d) Determine the conditional distributions. Use the conditionals to answer : (X, Y ) areindependent?e) Calculate E[XY ],...
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...
9. [15 Points) Let C be the boundary of the triangle with vertices (1, 1), (2,...
9. [15 Points) Let C be the boundary of the triangle with vertices (1, 1), (2, 3) and (2, 1), oriented positively i.e. counterclockwise). Let F be the vector field F(1, y) = (e* + y²)i + (ry + cos y)j. Compute the line integral F. dr. 10. (15 Points) Let S be the portion of the paraboloid z = 1-rº-ythat lies on and above the plane z = 0. S is oriented by the normal directed upwards. If F...
3. (2 Points) Let Q be the quadrilateral in the ry-plane with vertices (1, 0), (4,0), (0, 1), (0,4). Consider 1 dA...
3. (2 Points) Let Q be the quadrilateral in the ry-plane with vertices (1, 0), (4,0), (0, 1), (0,4). Consider 1 dA I+y Deda (a) Evaluate the integral using the normal ry-coordinates. (b) Consider the change of coordinates r = u-uv and y= uv. What is the image of Q under this change of coordinates?bi (c) Calculate the integral using the change of coordinates from the previous part. Change of Variables When working integrals, it is wise to choose a...
Use (part A) line integral directly then use (part B) Stokes' Theorem 10. Let C be the triangle from (0, 0,0) to (2, 0, 0) to (0, 2, 1) to (0, 0, 0) which lies in the plane z 2 -Зугі + 4zj + 6x k...
Use (part A) line integral directly then use (part B) Stokes'
Theorem
10. Let C be the triangle from (0, 0,0) to (2, 0, 0) to (0, 2, 1) to (0, 0, 0) which lies in the plane z 2 -Зугі + 4zj + 6x k, calculate | F . dr using Stokes's Theorem. If F(x, y, z) (b) 14 3 (c) 2 (d) 0 (e) None of these
10. Let C be the triangle from (0, 0,0) to (2,...
1. (a) A point is selected at random on the unit interval, dividing it into two pieces with total length 1. Find the pr...
1. (a) A point is selected at random on the unit interval, dividing it into two pieces with total length 1. Find the probability that the ratio of the length of the shorter piece to the length of the longer piece is less than 1/4 3 marks (b) Suppose X, and X2 are two iid normal N(μ, σ2) variables. Define Are random variables V and W independent? Mathematically justify your answer. 3 marks] (c) Let C denote the unit circle...
Let ? be the area in the ??-plane bounded by the sides of a triangle ???...
Let ? be the area in the ??-plane bounded by the sides of a triangle ??? with the vertices ? (1.1), ? (1.2). ? (2.2). Find the volume of the body you get when ? rotates the line ? = 10/3.
Let C be a triangle in the x-y
plane with vertices (x1,y1), (x2y2) and (x3,y3) arranged so that C
is positively-oriented.
Let C be a triangle in the xy-plane with vertices (x,y), (z2,p), and (z3,U3) arranged so that C is positively-oriented. a.) Sketch such a triangle and indicate its orientation. b.) Apply Green's Theorem to compute the area of the triangle as a (sum of) path integral(s) around the boundary. Get a formula for area in terms of the coordinates...
Let C be a triangle in the ry-plane with vertices (ıv). (2.92), and (T3, Vs) arranged so that C' is positively-oriented a.) Sketch such a triangle and indicate its orientation. b.) Apply Green's Theorem to compute the area of the triangle as a (sum of) path integral(s) around the boundary. Get a formula for area in terms of the coordinates zi and y. This formula is sometimes referred to as the Shoelace formula for area. c.) Use the formula you...
(1 point) Let T'be the region inside the triangle with vertices (0, 0), (3, 0) and (3, 2), and let f(x, y) be the function which is 0 outside of T and f(x, y)-38 SK y Ior (x, y) inside T. 3888 Then E(XY)- 4.8
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...
9. [15 Points) Let C be the boundary of the triangle with vertices (1, 1), (2, 3) and (2, 1), oriented positively i.e. counterclockwise). Let F be the vector field F(1, y) = (e* + y²)i + (ry + cos y)j. Compute the line integral F. dr. 10. (15 Points) Let S be the portion of the paraboloid z = 1-rº-ythat lies on and above the plane z = 0. S is oriented by the normal directed upwards. If F...
3. (2 Points) Let Q be the quadrilateral in the ry-plane with vertices (1, 0), (4,0), (0, 1), (0,4). Consider 1 dA I+y Deda (a) Evaluate the integral using the normal ry-coordinates. (b) Consider the change of coordinates r = u-uv and y= uv. What is the image of Q under this change of coordinates?bi (c) Calculate the integral using the change of coordinates from the previous part. Change of Variables When working integrals, it is wise to choose a...
Use (part A) line integral directly then use (part B) Stokes'
Theorem
10. Let C be the triangle from (0, 0,0) to (2, 0, 0) to (0, 2, 1) to (0, 0, 0) which lies in the plane z 2 -Зугі + 4zj + 6x k, calculate | F . dr using Stokes's Theorem. If F(x, y, z) (b) 14 3 (c) 2 (d) 0 (e) None of these
10. Let C be the triangle from (0, 0,0) to (2,...
1. (a) A point is selected at random on the unit interval, dividing it into two pieces with total length 1. Find the probability that the ratio of the length of the shorter piece to the length of the longer piece is less than 1/4 3 marks (b) Suppose X, and X2 are two iid normal N(μ, σ2) variables. Define Are random variables V and W independent? Mathematically justify your answer. 3 marks] (c) Let C denote the unit circle...
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