Heena Haldankar, LECTURER AND PROPRIETOR at HEENA HALDANKARS ACADEMY, Numerade Educator

09:37

Probability with Applications in Engineering, Science, and Technology

A large but sparsely populated county has two small hospitals, one at the south end of the county and the other at the north end. The south hospital's emergency room has 4 beds, whereas the north hospital's emergency room has only 3 beds. Let $X$ denote the number of south beds occupied at a particular time on a given day, and let $Y$ denote the number of north beds occupied at the same time on the same day. Suppose that the two rvs are independent, that the pmf of $X$ puts probability masses $.1, .2, . .3, .2,$ and .2 on the $x$ values $0,1,2,3,$ and $4,$ respectively, and that the $\mathrm{pmf}$ of $Y$ distributes probabilities $.1, .3, .4,$ and .2 on the $y$ values $0,1,2,$ and $3,$ respectively.
(a) Display the joint pmf of $X$ and $Y$ in a joint probability table.
(b) Compute $P(X \leq 1$ and $Y \leq 1$ ) by adding probabilities from the joint pmf, and verify that this equals the product of $P(X \leq 1)$ and $P(Y \leq 1)$.
(c) Express the event that the total number of beds occupied at the two hospitals combined is at most 1 in terms of $X$ and $Y,$ and then calculate this probability.
(d) What is the probability that at least one of the two hospitals has no beds occupied?

Chapter 4: Joint Probability Distributions and Their Applications

Section 1: Jointly Distributed Random Variables