2. On this problem, give exact answers Xing fractions. Let (X,Y) be a random point in the unit square, [0,1] × [0, 1], with the uniform probability. That is, X is a random number between 0 and 1, and Y is a random number between 0 and 1. Your overall task in this problem is to compute P(X < (3/4) given Y < (1/2)X). (a) Draw the Cartesian plane (make sure there is plenty of room in the first quadrant), and draw and label the axes. Sketch the sample space, [0, 1] × [0, 1], in the first quadrant. (b) Draw the graph of the line Y = (1/2)X in the sample space. Label the points where this line intersects the corners or sides of the unit square. (c) Compute P(Y < (1/2)X) by finding the area of the triangular region underneath the line from part (b) and dividing by the area of the sample space. (d) Draw the graph of the vertical line X = (3/4) in the sample space. Label the points where this line intersects the sides of the unit square, and the point where this line intersects the line Y = (1/2)X from part (b). (e) Sketch the region where X < (3/4) and Y < (1/2)X. Use shading or coloring. (f) Compute P(X < (3/4) and Y < (1/2)X) by finding the area of the triangular region from part (e) and dividing by the area of the sample space. (g) Use your answers to parts (f) and (c) to compute the conditional probability P(X < (3/4) given Y < : (1/2)X).

For the M/M/N/o system, the probability that an arrival will find all servers busy and will be forced to wait in queue is an important measure of performance of the M/M/N/∞ system. This probability is given by PQ N!(1 – p/N) | and is known as the Erlang C formula. Please derive the equation. What is the expected number of customers waiting in the queue (not in service)?

The complement of an event is all possible outcomes not in the event.  For example: If A = at least 3 girls are born to a family with 6 kids.  Then A' =  A compliment is what happens if there are not at least 3 girls = less than 3 girls born to the family.  A useful formula: P(A) + P(A' ) = 1 or P(A') = 1 - P(A).  If P(A) = 0.194, what is P(A')?

I roll a die with 10 faces numbered 0-9 and obtain a score X. What is E(X)? In order to obtain a random number between 0 and 99 I roll the die twice obtaining scores X and Y, then set Z = 10X + Y (i.e., the first roll determines the "tens", the second roll the "ones"). Is it true to say that E(Z) = 10E(X) + E(Y)?

Use the data in the table below, which shows the employment status of individuals in a particular town by age group. Part-time Unemployed Age Full-time 0-17 25 187 252 18-25 391 200 199 26-34 434 66 22 35-49 562 192 149 50+ 461 170 237 If a person is randomly chosen from the town's population, what is the probability that the person is under 18 or employed part-time?

Q.3. Compute the probability that: (a) their sum is odd; (b) their product is even. Three distinct integers are chosen at random from the first 15 positive integers.

For the random variables X,Y Cov(X,Y) = -0.9 if Z=3-X then what is Cov(Z,Y)=???

Just need help with part D please!