Let (X, Y ) has a uniform density in the unit circle, i.e., f(x, y) =...

Let (X, Y ) has a uniform density in the unit circle, i.e., f(x, y) = c, x2 + y 2 ≤ 1, for some constant c > 0.

• (a) Find E[X].

• (b) Find the conditional pdf of X given Y = y.

Solutions

Expert Solution

orchestra answered 1 year ago

Next > < Previous

Related Solutions

1. Let X be the uniform distribution on [-1, 1] and let Y be the uniform...

1. Let X be the uniform distribution on [-1, 1] and let Y be the uniform distribution on [-2,2]. a) what are the p.d.f.s of X and Y resp.? b) compute the means of X, Y. Can you use symmetry? c) compute the variance. Which variance is higher?

Let X, Y be independent random variables with X ∼ Uniform([1, 5]) and Y ∼ Uniform([2,...

Let X, Y be independent random variables with X ∼ Uniform([1, 5]) and Y ∼ Uniform([2, 4]). a) FindP(X<Y). b) FindP(X<Y|Y>3) c) FindP(√Y<X<Y).

The curried version of let f (x,y,z) = (x,(y,z)) is let f (x,(y,z)) = (x,(y,z)) Just...

The curried version of let f (x,y,z) = (x,(y,z)) is let f (x,(y,z)) = (x,(y,z)) Just f (because f is already curried) let f x y z = (x,(y,z)) let f x y z = x (y z)

Let X and Y have the following joint density function f(x,y)=k(1-y) , 0≤x≤y≤1. Find the value...

Let X and Y have the following joint density function f(x,y)=k(1-y) , 0≤x≤y≤1. Find the value of k that makes this a probability density function. Compute the probability that P(X≤3/4, Y≥1/2). Find E(X). Find E(X|Y=y).

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

5. Let X, Y and Z be sets. Let f : X ! Y and g...

5. Let X, Y and Z be sets. Let f : X ! Y and g : Y ! Z functions. (a) (3 Pts.) Show that if g f is an injective function, then f is an injective function. (b) (2 Pts.) Find examples of sets X, Y and Z and functions f : X ! Y and g : Y ! Z such that g f is injective but g is not injective. (c) (3 Pts.) Show that if...

Suppose that the random variables, ξ, η have joint uniform density f(x, y) = 2/9 in...

Suppose that the random variables, ξ, η have joint uniform density f(x, y) = 2/9 in the triangular region bounded by the lines x = -1 , y - -1 and y = 1 - x. a) Find the marginal densities f(x) =∫ 2/9 dy (limits, -1 to 1-x) and f(y) =∫ 2/9 dx (limits -1 to 1-y). Also show that f(x) f(y) ≠ f(x, y) so that ξ and η are not independent. b) Verify that μξ = ∫...

Let f(x, y) = − cos(x + y2 ) and let a be the point a...

Let f(x, y) = − cos(x + y2 ) and let a be the point a = ( π/2, 0). (a) Find the direction in which f increases most quickly at the point a. (b) Find the directional derivative Duf(a) of f at a in the direction u = (−5/13 , 12/13) . (c) Use Taylor’s formula to calculate a quadratic approximation to f at a.

Let f (x, y) = c, 0 ≤ y ≤ 4, y ≤ x ≤ y...

Let f (x, y) = c, 0 ≤ y ≤ 4, y ≤ x ≤ y + 1, be the joint pdf of X and Y. (a) (3 pts) Find c and sketch the region for which f (x, y) > 0. (b) (3 pts) Find fX(x), the marginal pdf of X. (c) (3 pts) Find fY(y), the marginal pdf of Y. (d) (3 pts) Find P(X ≤ 3 − Y). (e) (4 pts) E(X) and Var(X). (f) (4 pts) E(Y)...

Let f ( x , y ) = x^ 2 + y ^3 + sin ⁡...

Let f ( x , y ) = x^ 2 + y ^3 + sin ⁡ ( x ^2 + y ^3 ). Determine the line integral of f ( x , y ) with respect to arc length over the unit circle centered at the origin (0, 0).

Subjects