Mathematics 5010-2 and 5010-3
Solutions to
Quiz 4

1. The gross weekly sales at a certain restaurant is a normal random variable with mean $2200 and standard deviation $230. What is the probability that the weekly sales exceed $2000 in at least 2 of the next 3 weeks? State one (and only one) independence assumption that you are making.

   SOLUTION.
   Assuming that weekly sales are independent, the random number X in question is a Binomial with n=3 and p = 0.81 (from the fig. below). Thus,

   P{ X >= 2 }	=	P{ X=2 } + P{ X=3 }
   	=	(3 choose 2) 0.812 (1-0.81) + (3 choose 3) 0.81 3
   	=	0.90

   

2. Two balls are chosen at random with replacement from an urn consisting of 6 white balls and 7 red balls. Let X_i=1, if the ith ball selected is white, and X_i=0, if the ith ball selected is red.
   1. Compute the conditional pmf of X₁, given that X₂=0.
   2. Do the same thing as above, but assume that the balls are selected without replacement.

   SOLUTION.

   1. P{ X₁=0 | X₀=0 } = 7/13 and P{ X₁=1 | X₀=0 } = 6/13 .
   2. P{ X₁=0 | X₀=0 } = 6/12 =1/2 and P{ X₁=1 | X₀=0 } = 1/2.