A municipal bond service has three rating categories and

A municipal bond service has three rating categories ( , , and

). Suppose that in the past year, of the municipal bonds issued throughout the United Sates, 70% were rated , 20% were rated , and 10% were rated . Of the municipal bonds rated , 50% were issued by cities, 40% by suburbs, and 10% by rural areas. Of the municipal bonds rated , 60% were issued by cities, 20% by suburbs, and 20% by rural areas. Of the municipal bonds rated , 90% were issued by cities, 5% by suburbs, and 5% by rural areas.

a) Given that a new municipal bond is issued by a suburb, what is

    the probability it will receive an rating?

b) Given that a new municipal bond is rated , what is the

    probability it was issued by a city?

c) What proportion of municipal bonds is issued by suburbs?

Please include all steps with the answer

Solution

[the symbols for rating categories are not showing in my PC, hence i am assigning the first category as A,2nd one as B, third one as C...please match this A,B or C with your ratings]

Let, A be the event that rating A was issued. B be the event that rating B was issued. C be the event that rating C was issued.

T be the event that the bond was issued by city
S be the event that it was issued by suburb
R be the event that it was issued by rural

it is given that--

P[A]=0.7 P[B]=0.2    P[C]=0.1

P[T|A]=0.5    P[S|A]=0.4 P[R|A]=0.1

P[T|B]=0.6   P[S|B]=0.2     P[R|B]=0.2

P[T|C]=0.9 P[S|C]=0.05     P[R|C]=0.05

part b.

given the municipal bond is rated A, the probability that it was issued by a city is--

P[T|A]=0.5           .................... (i)

given the municipal bond is rated B, the probability that it was issued by a city is--

P[T|B]=0.6          .................... (ii)

given the municipal bond is rated C, the probability that it was issued by a city is--

P[T|C]=0.9          .................... (iii)

[please refer to which rating is mentioned in the question. if it is the first category rating then refer (i)

if it is the second category rating then refer (ii) and if it is the third one refer (iii) ]

part c

proportion of municipal bonds issued by suburbs----

P[S]=P[S|A]*P[A]+P[S|B]*P[B]+P[S|C]*P[C]    [ BY TOTAL PROBABILITY THEOREM]

      =0.4*0.7+0.2*0.2+0.05*0.1=0.325 [ANSWER]

part a

here BAYES\' THEOREM is applicable.

probability that it will receive an A rating given it is issued by suburb---

P[A|S]={P[S|A]*P[A]}/{P[S|A]*P[A]+P[S|B]*P[B]+P[S|C]*P[C]}={0.4*0.7}/0.325   [the denominator is obtained from part c]

        =0.28/0.325=0.86153 (approx) ..............(i)

probability that it will receive an B rating given it is issued by suburb---

P[B|S]={P[S|B]*P[B]}/{P[S|A]*P[A]+P[S|B]*P[B]+P[S|C]*P[C]}={0.2*0.2}/0.325   [the denominator is obtained from part c]

        =0.04/0.325=0.12307 (approx) ..............(ii)

probability that it will receive an C rating given it is issued by suburb---

P[C|S]={P[S|C]*P[C]}/{P[S|A]*P[A]+P[S|B]*P[B]+P[S|C]*P[C]}={0.05*0.1}/0.325   [the denominator is obtained from part c]

        =0.005/0.325=0.01538 (approx) ..............(iII)

[please refer to which rating is mentioned in the question. if it is the first category rating then refer (i)

if it is the second category rating then refer (ii) and if it is the third one refer (iii) ]