8.2: Conditional Expectations, Correlation Flashcards
What is an independent event? What are the conditions (2), and what happens if the conditions are not satisfied?
An independent event refers to the occurrence of one that has no influence on the occurrence of others.
It is expressed in terms of conditional probabilities and A and B are independent if and only if:
P(A|B) = P(A), or equivalently, P(B|A) = P(B)
If this condition is not satisfied, the events are dependent events.
How is unconditional probability interpreted using the total probability rule?
The unconditional probability of A is the probability-weighted sum of the conditional probabilities where B is a set of mutually exclusive and exhaustive events.
** see equation
How is conditional probability interpreted using the total probability rule?
Conditional expected values depend on the outcome of some other event.
What does a tree diagram exhibit?
A tree diagram is a general framework that shows the probabilities of various outcomes.
What does a tree diagram exhibit? What do the probabilities of the different branches sum up to?
A tree diagram is a general framework that shows the probabilities of various outcomes.
All the probabilities of the different branches sum up to 1.
Define covariance and how it is represented using a probability model.
Covariance is a measure of how two assets move together. It is the expected value of the product of the deviations of the two random variables from their respective expected values.
COV(Ri,Rj) = E{ [Ri - E(Ri)] [Rj - E(Rj)] }
What are the 3 properties of covariance?
Properties:
1. The covariance is a general representation of the same concept as the variance. The difference is that variance measures a single variable, while covariance measures how one variable moves with another variable.
- The covariance of a variable with itself is equal to its variance:
Cov(Ra,Ra) = Var(Ra)
- The covariance may range from negative infinity to positive infinity.
How is covariance represented using historical data?
COV(A,B) = sum of (return on A times mean return on A) times (return on B times mean return on B) over number of periods minus 1
What is the disadvantage to using covariance?
Covariance is difficult to interpret. Hence, using the correlation coefficient helps to see the relationship.
How is correlation different between a probability model vs. historical data?
Correlation can be forward-looking if it uses covariance from a probability model.
Correlation can be backward-looking if it uses covariance from historical data.
How is correlation coefficient represented?
Correlation = covariance over standard deviation of A times standard deviation of B
What are the 3 properties of correlation? Explain each.
Properties:
- Correlation measures the strength of the linear relationship between two random variables.
- Correlation has no units
- Correlation ranges from -1 to +1
if correlation = 1 the random variables have perfect positive correlation.
if correlation = -1 the random variables have perfect negative correlation.
if correlation = 0 there is no linear relationship between the variables.
Describe scatterplots. What’s on its x-axis and y-axis? What is its advantage?
Scatterplots are a method for displaying the relationship between two variables.
X-axis is one variable, y-axis is the other. Paired observation is plotted as a single point.
Advantage: can reveal non-linear relationships, which are not described by the correlation coefficient.
What’s important to note about causation in correlation? What is one way to test the causal relationship?
Causation is not implied from significant correlation. If it were, which variable is causing change in the other is not revealed by correlation. The natural of any causal relationship should be separately investigated or based on theory that can be subject to additional tests.
One way to test the causal relationship is by removing the outliers. If the correlation reduces significantly, further inquiry is necessary into whether the outliers provide information or are caused by noise in the data used.
Define spurious correlation. Explain with an example.
Spurious correlation refers to correlation that is either the result of chance or present due to changes in both variables over time that is caused by their association with a third variable.
For instance, correlation between the age of each year’s Miss America and the number of films Nicolas Cage appeared in the same year is 87%, which is random.
