6.6. A computer device can be either in a busy mode (state 1) processing a task, or in an idle 162 Probability and Statistics for Computer Scientists mode (state 2), when there are no tasks to process. Being in a busy mode, it can finish a task and enter an idle mode any minute with the probability 0.2. Thus, with the probability 0.8 it stays another minute in a busy mode. Being in an idle mode, it receives a new task any minute with the probability 0.1 and enters a busy mode. Thus, it stays another minute in an idle mode with the probability 0.9. The initial state X0 is idle. Let Xn be the state of the device after n minutes. (a) Find the distribution of X2. (b) Find the steady-state distribution of Xn

In the image attached you can find the Unit 7 homework.

We need to findt he missing measures of each figure.

Notice that the first figure is a rectangle, which means opposite sides are congruent so,

VY = 19

WX = 19

YX = 31

VW = 31

To find the diagonals we need to use Pythagorean's Theorem, where the diagonals are hypothenuses.

Also, , beacuse rectangles have congruent diagonals, which intercect equally.

That means,

Figure number two is also a rectangle.

If GH = 14, that means diagonal GE = 28, because diagonals intersect in equal parts.

Now, GF = 11, because rectangles have opposite sides congruent.

DF = 28, because in a reactangle, diagonals are congruent.

HF = 14, because its half of a diagonal.

To find side DG, we need to use Pythagorean's Theorem, where GE is hypothenuse

This figure is also a rectangle, which means all four interior angles are right, that is, equal to 90°, which means angle 11 and the 59° angle are complementary, so

Now, angles 11 and 4 are alternate interior angles which are congruent, because a rectangle has opposite congruent and parallel sides.

Which means , beacuse it's the complement for angle 4.

Now, , because it's a base angle of a isosceles triangle. Remember that in a rectangle, diagonals are congruent, and they intersect equally, which creates isosceles triangles.

, by interior angles theorem.

, by vertical angles theorem.

, by supplementary angles.

, by vertical angles theorem.

, by complementary angles.

, by alternate interior angles.

, by complementary angles.

, because it's one of the four interior angles of a rectangle, which by deifnition are equal to 90°.

, by alternate interior angles and by given., by complementary angles.

, by complementary angles.

, by interior angles theorem.

, by supplementary angles.

, by supplementary angles.

, by interior angles theorem, and by isosceles triangle theorem.

, by definition of rectangle.

, by interior angles theorem, and by isosceles triangle theorem.

, by complementary angles.

, by alternate interior angles.

The figure is a rectangle, which means its opposite sides are equal, so

Then, we replace this value in the expression of side WZ

Therefore, side WZ is 29 units long.

We know that the diagonals of a rectangle are congruent, so

Then,

Therefore, side PR is 73 units long.