A doctor's office has four telephone lines, each of which can be independently dialed by
any patient. Assume that, on any given business day, the probability that the first call is
received on line 1, 2, 3 and 4 are 0.2, 0.3, 0.2 and 0.3, respectively. And the first calls are
independent from one business day to the other. For n business days, let $Y_i$ be the number
of days on which the first call arrives on line $i$, $i=1,2,3,4$.
i. (2 pts) What is the distribution for the random vector ($Y_1, Y_2, Y_3, Y_4$)? Write the
name of the distribution with its parameters.
ii. (2 pts) What is the distribution for the random variable $Y_4$? Write the name of the
distribution with its parameters.
iii. (4 pts) Let n=10, find $P(Y_1=2, Y_2=4, Y_3=2)$
iv. (4 pts) Let n=8, write the expression for $P(Y_3=Y_4)$
v. (4 pts) Let n=10. Given that $Y_4$=2, compute the probability that $Y_1=2, Y_2=4, Y_3=2$.
