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# Probability matrix

#### QuantFFM

##### Member
Hello,

i have the following question:

1. The joint probability for two independent events is defined as: P(A and B)=P(A)xP(B)

2. Why is not P(High=20%)xP(Increase=40%)=P(Increase and High=10%).
Or should it be and only the example is wrong?

I wish all of you a nice Sunday evening.

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#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @QuantFFM Your first statement is correct as a condition for independence between the variables such that the failure of the condition applied to the matrix proves that GPD and credit spread are not independent. There are six "interior" probabilities and fully four of them, by my count, do not satisfy the condition for independence (although we only require one exception): joint P[GDP = i, CS = j] = P[GDP = i]*P[CS = j].

What is necessarily true, instead, is that joint P[GDP = i, CS = j] = P[CS = j | GDP = i ]*P[GDP = i]; i.e., Joint Pr[GDP, S] = Conditional Pr [CS | GDP] * Unconditional Pr [GDP]. So what's true per your probability matrix is:
• Joint P[high GDP, CS increase] = 10% = 20% * 10%/20% = 20% * 50%; where 20% is Prob [GDP = high] and 10%/20% = 50% is the conditional Prob [CS increase | GDP high]. If GPD and spread were independent, then conditional Prob [CS increase| GDP high] would necessarily equal Prob[CS increase], because CS increase would not depend on GDP, and then the always-true P[GDP = i, CS = j] = P[CS = j | GDP = i ]*P[GDP = i] would reduce to true-only-if-independent P[GDP = i, CS = j] = P[CS = j]*P[GDP = i] because only if they are independent can we replace P[CS = j | GDP = i ] with P[CS = j] ... but these random variable are not independent
• Working the other direction, it must be true that joint P[high GDP, CS increase] = 10% = 40%*10/40% = unconditional Prob[CS increase] * conditional Prob[GDP high | CS increase]. I hope that helps!

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#### Bansari

##### New Member
4. Your firm is testing a new quantitative strategy. The analyst who developed the
strategy claims that there is a 55% probability that the strategy will generate
positive returns on any given day. After 20 days the strategy has generated a
profit only 10 times. What is the probability that the analyst is right and the
actual probability of positive returns for the strategy is 55%? Assume that there
are only two possible states of the world: Either the analyst is correct, or there
the strategy is equally likely to gain or lose money on any given day. Your prior
assumption was that these two states of the world were equally likely.

5. Your firm has created two equity baskets. One is procyclical, and the other is
countercyclical. The procyclical basket has a 75% probability of being up in
years when the economy is up, and a 25% probability of being up when the
economy is down or flat. The probability of the economy being down or flat in
any given year is only 20%. Given that the procyclical index is up, what is the
probability that the economy is also up?

Can anybody help me with the solution of the above questions mentioned in practice. @David Harper CFA FRM

#### Nicole Seaman

Staff member
Subscriber
4. Your firm is testing a new quantitative strategy. The analyst who developed the strategy claims that there is a 55% probability that the strategy will generate positive returns on any given day. After 20 days the strategy has generated a profit only 10 times. What is the probability that the analyst is right and the actual probability of positive returns for the strategy is 55%? Assume that there are only two possible states of the world: Either the analyst is correct, or there the strategy is equally likely to gain or lose money on any given day. Your prior assumption was that these two states of the world were equally likely.

5. Your firm has created two equity baskets. One is procyclical, and the other is countercyclical. The procyclical basket has a 75% probability of being up in years when the economy is up, and a 25% probability of being up when the economy is down or flat. The probability of the economy being down or flat in any given year is only 20%. Given that the procyclical index is up, what is the probability that the economy is also up?

Can anybody help me with the solution of the above questions mentioned in practice. @David Harper CFA FRM
Hello @Bansari

Can you provide the source of these questions please? This is helpful when David answers questions in the forum, and it can also help us to tag the questions correctly so other members can reference them.

Thank you,

Nicole

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
@Nicole Seaman These questions are from Miller's own EOC Q&A, and he gives the answers in the book. I am currently updating the Miller XLS (as you know in anticipation for Deepa's revision; these are going to be sweet!). @Bansari must not have access to the entire book?

@Nicole Seaman can we look at the viability of adding Miller's EOC to our revised Miller Ch2, 3, 4, 6, 7 (maybe Deepa can include in her work)? ... because I would like to also capture them in the associated learning XLS. Thanks!

#### Nicole Seaman

Staff member
Subscriber
@Nicole Seaman These questions are from Miller's own EOC Q&A, and he gives the answers in the book. I am currently updating the Miller XLS (as you know in anticipation for Deepa's revision; these are going to be sweet!). @Bansari must not have access to the entire book?

@Nicole Seaman can we look at the viability of adding Miller's EOC to our revised Miller Ch2, 3, 4, 6, 7 (maybe Deepa can include in her work)? ... because I would like to also capture them in the associated learning XLS. Thanks!
@David Harper CFA FRM

Thank you for letting me know the source of these questions! Yes, I will definitely take a look at the EOC Q&A and add them to our updated Miller study notes! I've included it in my task list for the Miller revision.

#### Bansari

##### New Member
@Nicole Seaman @David Harper CFA FRM these questions are from FRM material provided from the GAARP institute. (Textbooks). Can you also tell me where can we get the solutions for all the Material question to confirm with my calculation and method and answer.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
@Bansari thank you. The answers are in Miller's (full) book. As mentioned, if you can wait, we have a revision of the Miller coming that will include them (Deepa is working on T8.Bodie now, then Miller is next, so it will be within the next month). I'm currently prepping the (extensive) XLS that supports those notes with Exhibits, and I definitely want to realize these Q&A in our (hopefully more intuitive) XLS format, because Bayes really wants graphical explanations IMO; my Miller XLS will be done this week, so i can share the answers (at least to these questions) when i've done that, if it's helpful. Otherwise, there are literally at the end of Miller's book, I guess GARPs readings extract the chapters but not the answers at the end?

#### QuantFFM

##### Member
Hi @QuantFFM Your first statement is correct as a condition for independence between the variables such that the failure of the condition applied to the matrix proves that GPD and credit spread are not independent. There are six "interior" probabilities and fully four of them, by my count, do not satisfy the condition for independence (although we only require one exception): joint P[GDP = i, CS = j] = P[GDP = i]*P[CS = j].

What is necessarily true, instead, is that joint P[GDP = i, CS = j] = P[CS = j | GDP = i ]*P[GDP = i]; i.e., Joint Pr[GDP, S] = Conditional Pr [CS | GDP] * Unconditional Pr [GDP]. So what's true per your probability matrix is:
• Joint P[high GDP, CS increase] = 10% = 20% * 10%/20% = 20% * 50%; where 20% is Prob [GDP = high] and 10%/20% = 50% is the conditional Prob [CS increase | GDP high]. If GPD and spread were independent, then conditional Prob [CS increase| GDP high] would necessarily equal Prob[CS increase], because CS increase would not depend on GDP, and then the always-true P[GDP = i, CS = j] = P[CS = j | GDP = i ]*P[GDP = i] would reduce to true-only-if-independent P[GDP = i, CS = j] = P[CS = j]*P[GDP = i] because only if they are independent can we replace P[CS = j | GDP = i ] with P[CS = j] ... but these random variable are not independent
• Working the other direction, it must be true that joint P[high GDP, CS increase] = 10% = 40%*10/40% = unconditional Prob[CS increase] * conditional Prob[GDP high | CS increase]. I hope that helps!
THX a lot. The forum is really helpful.