Let X and Y denote the values of two stocks at the end of a five-year period. X is uniformly distributed on the interval (0, 12). Given X = x, Y is uniformly distributed on the interval (0, x). Determine Cov(X, Y) according to this model.

Cov (X,Y) = 6

Step-by-step explanation:

hello,

Cov(X,Y) = E(XY) - E(X)E(Y)

we must  first find E(XY), E(X), and E(Y).

since X is uniformly distributed on the interval (0,12), then E(X) = 6.

next we find the joint density f(x,y) using the formula

this is because f is uniformly distributed on the the interval (0,12)

also since the conditional probability density of Y given X=x, is  uniformly distributed on the interval [0,x], then

 for 0≤y≤x≤12

thus

.

hence,

also,

thus Cov(X,Y) = E(XY) - E(X)E(Y)

                      =  24 - (6)(3)

                      =     6

b. an equilateral triangle inscribed in a circle

step-by-step explanation:

we are given the figure of two circles having a triangle inside.

now, we know that,

"when a figure is lying inside an another figure, then the outside figure is known as circumscribed figure and the figure lying inside is referred to as the inscribed figure".

as we see that, the circles are containing the triangle completely.

thus, the circles are the circumscribed figures and the triangle is the inscribed figure.

hence, we get, 'an equilateral triangle is inscribed in a circle'.