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# P1.T2.204. Joint, marginal, and conditional probability functions (Stock & Watson)

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
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AIM: Describe Joint, marginal, and conditional probability functions

Questions:

204.1. X and Y are discrete random variables with the following joint distribution; e.g., Pr (X = 4, Y = 30) = 0.07. What is the standard deviation of the conditional probability Pr (Y | X = 7)?

a. 10.3
b. 14.7
c. 21.2
d. 29.4

204.2. Sally's commute (C) is either long (L) or short (S). While commuting, it either rains (R = Y) or it does not (R = N). Today, the marginal (aka, unconditional) probability of no rain is 75%; P(R = N) = 75%. The joint probability of rain and a short commute is 10%; i.e., P(R = Y, C = S) = 10%. What is the probability of a short commute conditional on it being rainy, P (C = S | R = Y)?

a. 10%
b. 25%
c. 40%
d. 68%

204.3. Economists predict the economy has a 40% of experiencing a recession in 2012; marginal P(R) = 40% and therefore the marginal probability of no recession P(R') = 60%. Let P(S) be the probability the S&P 500 index ends the year above 1400, such that P(S') is the probability the index does not end the year above 1400. If there is a recession, the probability of the index ending the year above 1400 is only 30%; P(S|R) = 30%. If there is not a recession, the probability of the index ending above 1400 is 50%; P(S|R') = 50%. Bayes' Theorem tells us that the conditional probability, P(R|S), is equal to the joint probability P(R,S) divided by the marginal probability, P(S). At the end of the year, the index does end above 1400, such that we observe (S) not (S'). What is the probability of a recession conditional on the the index ending above 1400; i.e., P(R|S)?

a. 12.0%
b. 28.6%
c. 40.0%
d. 42.0%