Suppose that (X, Y) is uniform on the set {(x1, y1),...,(xn, yn)} where the x1,..., xn are distinct values and the y1,..., yn are distinct values. (a) Prove that X is uniformly distributed on x1,..., xn, with mean given by x¯ = n−1 3n i=1 xi and variance given by sˆ2 X = n−1 3n i=1 (xi − ¯x) 2 . (b) Prove that the correlation coefficient between X and Y is given by rXY = 3n i=1 (xi − ¯x) (yi − ¯y) T3n i=1 (xi − ¯x) 2 T3n i=1 (yi − ¯y) 2 = sˆXY sˆX sˆY where sˆXY = n−1 3n i=1 (xi − ¯x) (yi − ¯y). The value sˆXY is referred to as the sample covariance and rXY is referred to as the sample correlation coefficient when the values (x1, y1),...,(xn, yn) are an observed sample from some bivariate distribution. (c) Argue that rXY is also the correlation coefficient between X and Y when we drop the assumption of distinctness for the xi and yi . (d) Prove that −1 ≤ rXY ≤ 1 and state the conditions under which rXY = ±1

answer: ratio of volume of cone (with 3 times the height and equal diameter) to volume of cylinder is 2 to 1. the value of π is not used.

step-by-step explanation:

radius is diameter/2 = 4.

area of circle is π r^2.

volume of cylinder is height × area of base = hπr^2 = 3(4^2) π = 48π cubic inches.

not knowing the formula for volume of cone, i calculate it using calculus (which is for calculating things).

volume of cone is integral 0 to h area(y) dy. area(y) is π r(y)^2. r(y) runs r to zero as y runs zero to h, r(0) = r, r(h) = 0.

r(y) = r - r/h y

area(y) = π (r - r/h y)^2 = π (r^2 - 2r^2y/h + r^2y^2/h^2) = π r^2(1-2y/h+y^2/h^2)

antiderivative area(y) = π r^2 (y-y^2/h+y^3/(3h^2))

integral 0 to h = π r^2(h-h^2/h+h^3/(3h^2) - (0)) = π r^2(h - h + h/3) = π r^2(h/3) = π/3 r^2 h

volume of this cone is π/3(4^2)18 cubic inches.

= 6 16 π = 96π cubic inches.

cone has twice the volume, 96/48.

cone height is 6 times the cylinder.

ratio of volume formula: π r^2 cancels.

cone over cylinder ratio is 6h/3 to h, 2 to 1.

hello,

as we don't know if g(x) is continue,

answer b

f(x)=2x²-8x+1=2(x²-4x+4)+1-8=2(x-2)²-7

vertex is (2,-7) ( a minimum since coefficient of x² is > 0)

the answer is c. ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀