Learning Module 4 Probability Trees and Conditional Expectations

You have a discrete random variable X with possible outcomes 10 (probability 0.2), 8 (probability 0.5), and 5 (probability 0.3). What is E(X)?

Given the same distribution as Q1 (10 with p=0.2, 8 with p=0.5, 5 with p=0.3), what is Var(X)?

BankCorp faces two interest-rate scenarios: declining (probability 0.6) and stable (probability 0.4). Under declining rates, EPS is 2.60 with conditional prob 0.25 and 2.45 with prob 0.75. Under stable rates, EPS is 2.20 with prob 0.60 and 2.00 with prob 0.40. What is the unconditional probability that EPS = 2.45?

Using the BankCorp example from Q3, what is E(EPS | declining interest rate)?

Given same BankCorp example, compute unconditional E(EPS) using conditional expectations: E = 0.6*E(EPS|declining) + 0.4*E(EPS|stable). If E(EPS|declining)=2.4875 and E(EPS|stable)=2.12, what is E(EPS)?

A defaulted bond recovery problem: Scenario A (prob 0.75) gives $0.90 with prob 0.45 or $0.80 with prob 0.55. Scenario B (prob 0.25) gives $0.50 with prob 0.85 or $0.40 with prob 0.15. What is the unconditional probability of recovering $0.50?

You observe a new piece of information I. Priors: P(A)=0.45, P(B)=0.30, P(C)=0.25. Likelihoods: P(I|A)=0.75, P(I|B)=0.20, P(I|C)=0.05. What is P(I) (the unconditional probability of I)?

Using the priors and likelihoods from Q7, what is the posterior P(A | I)?

In the DriveMed example, after observing the firm expands (information), the posterior probabilities for EPS outcomes sums to 1. Which of these is an important consistency check after applying Bayes' formula?

A credit model gives prior P(repay)=0.90, P(good report)=0.80, and P(good report | repay)=0.85. Using Bayes' formula, what is P(repay | good report)?

When constructing a probability tree, how do you compute the joint probability at a terminal node?

You have two mutually exclusive scenarios S1 and S2 with P(S1)=0.7 and P(S2)=0.3. If E(X | S1) = 100 and E(X | S2) = 50, what is E(X)?

In Example 3 (BankCorp operating costs), Yhat = 12.5 + 0.65X representing expected operating costs (millions) given number of branches X. If X=100, what is the conditional expected operating cost?

If scenario probabilities change (e.g., target return increases), how does target downside semideviation behave if more observations fall below the target?

Which formula shows Bayes' theorem?

You use Bayes' formula to update P(non-survivor | fail test) given P(fail|non-survivor)=0.90, P(non-survivor)=0.40, and P(fail)=0.45. What is P(non-survivor | fail)?

In a probability tree with two sequential binary events, each branch labeled with its conditional probability, how many terminal nodes are there?

You have a prior P(default)=0.20. Your model rates 70% as 'good'; of the bonds that defaulted, 50% had 'good' rating. What is P(default | good rating)?

Which of the following is a correct statement about conditional expectation E(X | S)?

A probability tree with three branches at first node (S1,S2,S3) where probabilities are 0.2, 0.5, 0.3 respectively. If E(X|S1)=10, E(X|S2)=20, E(X|S3)=15, compute unconditional E(X).

If a posterior probability of an event increases compared to its prior after observing information, what can you say about P(info | event) relative to P(info)?

Which of the following is NOT an input required to compute a posterior using Bayes' formula?

You build a probability tree where a scenario S has prior 0.4. Under S, two outcomes X1 and X2 have conditional probabilities 0.3 and 0.7. What is joint probability of S and X2?

Which expression is the total probability rule for an event A across exhaustive scenarios S1..Sn?

An analyst says: 'If P(Info|Event) is very small, Bayes will always decrease the posterior compared to prior.' Is this always true?

You have a forecast tree: scenario probabilities 0.5 and 0.5. Under first scenario expected payoff = 10 with SD 2; under second expected payoff = 6 with SD 1. If you must report unconditional expected payoff and you know scenarios are independent, what is unconditional expectation?

When calculating conditional variance Var(X | S), which definition is used?

If Bayes' formula yields P(Event | Info) = 0.03 from a prior of 0.25 after observing Info, how did Info affect belief?

Suppose an analyst sets up three exhaustive events for a firm's performance. After observing Info, posteriors are computed as 0.60, 0.30, and 0.10. What property should these posteriors satisfy?

In the BankCorp EPS example, if interest rates are known to be stable (S observed), which expectation should be used to update EPS forecast?

Which of the following best describes a 'likelihood' in Bayesian updating?

You want to check consistency of probabilities derived from a probability tree. What must hold for all terminal nodes?

If Event A and Event B are independent, which of the following is true?

You observe Info with P(Info)=0.32, and you know P(Event)=0.2 and P(Info|Event)=0.6. What is posterior P(Event|Info)?

Which statement about conditional expectation is correct in investment context?

A manager claims that Bayes' formula is only useful when priors are precise. What is the chapter's perspective?

Which of the following is an advantage of probability-tree modeling in investments?

Which calculation checks that conditional probabilities in a tree are consistent with unconditional probabilities provided earlier?

You have two scenarios with prior 0.4 and 0.6. Under scenario1 expected payoff 50, variance 25; under scenario2 expected payoff 30, variance 9. If you want the unconditional variance of payoff, which components must be included?

An analyst believes a stock will beat consensus with prior 0.45, meet 0.30, fall short 0.25. Observed Info is capacity expansion. Likelihoods: 0.75, 0.20, 0.05 respectively. Which posterior is largest?

Which of these is a direct application of Bayes' formula in investment practice described in the chapter?

Which of the following steps is NOT part of the standard six-step hypothesis testing process outlined in the chapter?

In Bayes' example with stocks and >10% returns, there are 100 tech firms with 60 having >10% returns and 400 non-tech with 100 >10% returns. What is P(tech | R>10%)?

Which method would you use to compute the expected operating cost if you know growth probabilities and conditional branches for branch counts?

When applying Bayes in credit scoring, which numbers correspond to 'likelihoods'?

Which is true about Bayes' formula in investment settings?

An investor uses a probability tree to forecast returns under three macro scenarios. She observes which macro scenario occurred mid-year. Which concept allows her to revise the portfolio forecast immediately?

Which of the following best explains why probability trees aid communication in investment committees?