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?
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 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).
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 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 = ( 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 + 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 ( 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).