An understanding of imaginary numbers

1. When solving a quadratic equation using the quadratic formula, it is possible for the b2 - 4ac term inside the square root (the discriminant) to be negative, thus forcing us to take the square root of a negative number. The solutions to the equation will then be complex numbers (i.e., involve the imaginary unit i). Question: In the real world, where might these so-called imaginary numbers be used? 2. When using a formula, we often know the value of one variable to a greater degree of accuracy than we know the others. In your opinion, what affect, if any, does it make on our use of a formula if we know the value of one variable to a greater degree of accuracy than another?

Jointly Distributed Random Variables

10. Annie and Alvie have agreed to meet between 5:00 pm and 6:00 pm for dinner at a local health food restaurant. Let X=Annie's arrival time and Y=Alvie's arrival time. Suppose X and Y are independent with each uniformly distributed on the interval [5, 6]. a. What is the joint pdf of X and Y? b. What is the probability that they both arrive between 5:15 and 5:45? c. If the first one to arrive will wait only 10 minutes before leaving to eat elsewhere, what is the probability that they have dinner at the health-food restaurant? [Hint: The event of interest is A={(x,y): | x-y | ≤ 1/6}.] (Please use the given hint. Thanks.)

Jointly Distributed Random Variables

12. Two components of a minicomputer have the following joint pdf for their useful life times X and Y: (see attachment) a. What is the probability that the lifetime X of the first component exceeds 3? b. What are the marginal pdf's of X and Y? Are the two lifetimes independent? Explain. c. What is the probability that the lifetime of at least one component exceeds 3?

Compute the expected revenue ...

26. Consider a small ferry that can accomodate cars and buses. The toll for cars is $3, and the toll for buses is $10. Let X and Y denote the number of cars and buses, respectively, carried on a single trip. Suppose the joint distribution of X and Yis as given in the table in Exercise 7 (see attachment). Compute the expected revenue from a single trip.

Compute the covariance

Please assist me with the attached problems. Thanks. 30. a. Compute the covariance for X and Y in Exercise 22 (see attachment) b. Compute ρ for X and Y in the same exercise.