6. Pick a uniformly chosen random point (X, Y ) inside a unit square [0, 1]×[0,...

6. Pick a uniformly chosen random point (X, Y ) inside a unit square [0, 1]×[0, 1], and let M = min(X, Y ). Find the probability that M < 0.3.

Solutions

Expert Solution

orchestra answered 1 year ago

Next > < Previous

Related Solutions

A point (a, b) is distributed uniformly in the square 0<x<1, 0<y<1. Let S(a, b) be...

A point (a, b) is distributed uniformly in the square 0<x<1, 0<y<1. Let S(a, b) be the area of a rectangle with sides a and b. Find P{1/4 < S(a, b) < 1/3}

Break a stick of unit length at a uniformly chosen random point. Then take the shorted...

Break a stick of unit length at a uniformly chosen random point. Then take the shorted of the two pieces and break it again in two pieces at a uniformly chosen random point. Let X denote the shortest of the final three pieces. Find the density of X.

Choose a point at random from the unit square [0, 1] × [0, 1]. We also...

Choose a point at random from the unit square [0, 1] × [0, 1]. We also choose the second random point, independent of the first, uniformly on the line segment between (0, 0) and (1, 0). The random variable A is the area of a triangle with its corners at (0, 0) and the two selected points. Find the probability density function (pdf) of A.

A point is chosen uniformly at random from a disk of radius 1, centered at the...

A point is chosen uniformly at random from a disk of radius 1, centered at the origin. Let R be the distance of the point from the origin, and Θ the angle, measured in radians, counterclockwise with respect to the x-axis, of the line connecting the origin to the point. 1. Find the joint distribution function of (R,Θ); i.e. find F(r,θ) = P(R ≤ r, Θ ≤ θ). 2. Are R and Θ independent? Explain your answer.

Let X and Y be independent and uniformly distributed random variables on [0, 1]. Find the...

Let X and Y be independent and uniformly distributed random variables on [0, 1]. Find the cumulative distribution and probability density function of Z = X + Y.

Let X and Y be uniformly distributed independent random variables on [0, 1]. a) Compute the...

Let X and Y be uniformly distributed independent random variables on [0, 1]. a) Compute the expected value E(XY ). b) What is the probability density function fZ(z) of Z = XY ? Hint: First compute the cumulative distribution function FZ(z) = P(Z ≤ z) using a double integral, and then differentiate in z. c) Use your answer to b) to compute E(Z). Compare it with your answer to a).

1. Let X and Y be independent U[0, 1] random variables, so that the point (X,...

1. Let X and Y be independent U[0, 1] random variables, so that the point (X, Y) is uniformly distributed in the unit square. Let T = X + Y. (a) Find P( 2Y < X ). (b). Find the CDF F(t) of T (for all real numbers t). HINT: For any number t, F(t) = P ( X <= t) is just the area of a part of the unit square. (c). Find the density f(t). REMARK: For a...

Let X and Y be random variables. Suppose P(X = 0, Y = 0) = .1,...

Let X and Y be random variables. Suppose P(X = 0, Y = 0) = .1, P(X = 1, Y = 0) = .3, P(X = 2, Y = 0) = .2 P(X = 0, Y = 1) = .2, P(X = 1, Y = 1) = .2, P(X = 2, Y = 1) = 0. a. Determine E(X) and E(Y ). b. Find Cov(X, Y ) c. Find Cov(2X + 3Y, Y ).

Pick an arbitrary consumption bundle (x, y) ∈ R2+ (meaning x ≥ 0, y ≥ 0)....

Pick an arbitrary consumption bundle (x, y) ∈ R2+ (meaning x ≥ 0, y ≥ 0). Draw or describe the better than set (upper contour set), the worse than set (lower contour set) and the indifference set in a graph for the following situations: • I like consuming x, and the more the better. I am completely indifferent to the amount of y that I consume. • As long as my consumption of x is less than x∗, the more...

Let X and Y be uniform random variables on [0, 1]. If X and Y are...

Let X and Y be uniform random variables on [0, 1]. If X and Y are independent, find the probability distribution function of X + Y

Subjects