Exercise 3: Suppose that in an electric display sign there are three light bulbs in the first row and four light bulbs in the second row. Let X denote the number of bulbs in the first row that will be burned out at a specified time t, and let Y denote the mumber of bulbs in the second row that will be burned out at the same time t. Suppose that the joint p.f of X and Y is as specified in the following table: 2 3 4 0 0.08 0.07 0.06 0.01 0.01 0.06 0.10 0.12 0.05 0.02 2 0.05 0.06 0.09 0.04 0.03 3 0.02 0.03 0.03 0.03 0.04 Determine each of the following probabilities: 1. Pr(X = 2) 2. Pr(Y 2 2)

A service station has both self-service and full-service islands. On each island, there is a single regular unleaded pump with two hoses. Let X denote the number of hoses being used on the self-service island at a particular time, and let Y denote the number of hoses on the full-service island in use at that time. The joint pmf of X and Y appears in the accompanying tabulation.   y p(x, y)   0 1 2 x 0 0.10 0.05 0.01 1 0.07 0.20 0.08 2 0.05 0.14 0.30 (d) Compute the marginal pmf of X. x 0 1 2 pX(x)       Compute the marginal pmf of Y. y 0 1 2 pY(y)       Using pX(x), what is P(X ≤ 1)? P(X ≤ 1) =

The scenario relates to a waiting line at the Dipping star at Six Flags. For the purposes of the next two questions, assume that the Wicked Twister, which was permanently closed in September 2021, has since relocated and re-opened at a different venue in California.   Suppose that a waiting line at the Dipping Star at Six Flags follows M/M/1 queue, with unlimited arrivals, and FIFO.  Assume that on a busy holiday, visitors to the Dipping Star purchase tickets in person every five minutes on average.  Also, assume that the employees at the ticket counters on average serve fifteen customers every hour.  What is the expected average time a customer waits in line before it is their turn to purchase tickets?  a. 2 minutes b. 3.3 minutes c. 8 minutes d. 16 minutes e. 26.6 minutes On average, how many customers are waiting in line? a. 1.5 b. 1.9 c. 3.2 d. 4.1 e. 3.25 students

Suppose ū = (1,-5, 1) and ī = (0, –5, 3). Then: u + v= и — 0% U - u= 5u= 2u – T0=