Let Y1 and Y2 denote the proportions of time (out of one workday) during which employees I and II, respectively, perform their assigned tasks. The joint relative frequency behavior of Y1 and Y2 is modeled by the density function. f (y 1,y2)=y 1+y 2 o<=y 1<=1, 0<=y2<=1(0 elsewhere)

a. Find P (Y1< 1/2,y2>1/4)
b. Find P(Y 1+Y2<=1)
Are Y1 and Y2 independent?

(a) The region Y₁ < 1/2 and Y₂ > 1/4 corresponds to the rectangle,

{(y₁, y₂) : 0 ≤ y₁ < 1/2 and 1/4 < y₂ ≤ 1}

Integrate the joint density over this region:

(b) The line Y₁ + Y₂ = 1 cuts the support in half into a triangular region,

{(y₁, y₂) : 0 ≤ y₁ < 1 and 0 < y₂ ≤ 1 - y₁}

Integrate to get the probability:

Y₁ and Y₂ are not independent because

P(Y₁ = y₁, Y₂ = y₂) ≠ P(Y₁ = y₁) P(Y₂ = y₂)

To see this, compute the marginal densities of Y₁ and Y₂.

but this clearly does not match the joint density.

answer: 457.14

step-by-step explanation:

205 - 45 = 160

160 / .35 = 457.142857