FRM distributions

What does Bernoulli's distribution represent?

A single trial with E[X] = p and Var = p(1-p), often used for binary outcomes like Default/No Default.

What is the key difference between Binomial and Bernoulli distributions?

Binomial distribution is for n trials, while Bernoulli is for a single trial.

When is the Poisson distribution used?

For modeling rare events in a specific time frame, with E[X] = λ and Var = λ.

How does Poisson scaling work for monthly calculations?

If λ is annual, divide by 12 to get monthly rates.

What is the criterion for applying the Poisson approximation?

Applicable when n > 100 and np < 10.

Describe the Normal distribution in terms of symmetry and skewness.

It is symmetrical with skewness = 0, kurtosis = 3, and excess kurtosis = 0.

What type of data does the Lognormal distribution model?

Used to model asset prices, it is positively skewed and bounded at 0.

What is the formula for the mean and variance of a Lognormal distribution?

Mean = e^{μ + 0.5σ^2}; Variance = [e^{σ^2} - 1] ⋅ e^{2μ + σ^2}.

When do you use the Student's t distribution?

For small samples (n < 30) or when variance is unknown.

What is the relationship of the t-distribution to the Normal distribution as degrees of freedom increase?

As df → ∞, the t-distribution approaches the Normal distribution.

What does the Chi-Square distribution calculate?

It is the sum of n squared standard normals, generally used to test variance.

What is the F-Distribution used for?

To test the equality of two variances, with the larger variance in the numerator.

What does the Central Limit Theorem state?

The distribution of sample means approaches Normal as n increases, even if the underlying data is skewed.

What transformations can be done from Normal to other distributions?

Normal can be transformed to Lognormal (e^X) or Chi-Square (Z^2).

What distinguishes discrete and continuous decision trees?

Discrete decision trees count fixed trials (Binomial), while continuous ones handle prices/variables (Lognormal) and returns (Normal).

What do Leptokurtic and Platykurtic distributions indicate about Value at Risk (VaR)?

Leptokurtic (Kurt > 3) underestimates VaR, while Platykurtic (Kurt < 3) overestimates VaR.

What are the critical values for a Z-Table at various confidence levels?

95% (1-tail) = 1.645, 99% (1-tail) = 2.33, 95% (2-tail) = 1.96.

How do significance levels relate to confidence levels?

A 5% significance level corresponds to a 95% confidence level.