A restaurant needs a staff of 3 waiters and 2 chefs to be pr

A restaurant needs a staff of 3 waiters and 2 chefs to be properly staffed. The joint probability model for the number of waiters (X) and chefs (Y) that show up on any given day is given below.

e) What is the expected total number of staff (waiters and chefs) that will show up on any given day?  

f) What is the probability that three waiters will show up on any given day?  

g) What is the probability that two chefs will show up on any given day?  

h) X and Y are independent.

False or True?

Solution

e) Expected total no.of staff is given by E(X)(Y) = (0x1)x0.02 + (0x2)x0.02 + (1x0)x0.03 + (1x1)x0.03 + (1x2)x0.01 + (2x0)x0.02 + (2x1)x0.04 + (2x2)x0.05 + (3x0)x0.04 + (3x1)x0.04 + (3x2)x0.68 = 4.53

f) Probability that three waiters will show up on any given day is when (X=3) i.e. 0.04+0.04+0.68 =0.76

g) Probability that two chefs will show up on any given day is when(Y=2) i.e. 0.01+0.01+0.05+0.68 =0.75

h) False,as can be seen from the fact that P(X = 1, Y = 1) not equals P(X = 1)P(Y = 1). (The left side is 0.03, but the right side is 0.03 × 0.02 = 0.0006.)