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Bernoulli Distribution


  • Assumptions
    • The outcome of an experiment can either be success (i.e., 1) and failure (i.e., 0)
  • Pr(X=1) = p, Pr(X=0) = 1-p, or
  • The expected value E[X] of the event is equal to the probability of success(p)
    • E[X] = p
  • The variance of the event is the product of the probability of success and probability of failure:
    • Var(X) = p(1-p)
  • Assumptions:
    • There are n trials
    • Each trial has two possible outcomes, “success” or “failure”
    • The probability of success p is the same for each trial
    • Each trial is independent
  • If we take n Bernoulli trials, and say out of those n trials we have total number of “x” successes, then the probability of such an event can be given as:
  • The expected number of successes E[X] = n * p  
  • The variance of number of successes Var (X) = n * p * (1-p)


There are 10 bonds in a credit default swap basket; the probability of default for each of the bonds is 5%. The probability of any one of the bond defaulting is completely independent of what happens to the other bonds in the basket. What is the probability exactly one bond default?

  1. 5%
  2. 50%
  3. 32%
  4. 3%

One particular bond defaults and other nine do not with the probability 0.05* (0.95)^9 can happen in 10 different ways
= 10 * 0.05^1* (0.95)^9 = 32%



Company ABC was incorporated on January 1, 2004. it has expected annual default rate of 10%. Assuming a constant quarterly default rate, what is the probability that company ABC will not have defaulted by April 1, 2004?


Let the probability of not defaulting in 1 quarter is (nd). Then the probability of not defaulting for a full year is (nd)4. This implies that the probability of defaulting within 1 year time is {1-(nd)4}, which is given as 10%.
1-(nd)4 = 0.1 which implies
(nd) = 0.91/4
= 97.4%



A corporate bond will mature in 3 years. The marginal probability of default in year one is 0.03%. The marginal probability of default in year 2 is 0.04%. The marginal probability of default in year 3 is 0.06%. What is the cumulative probability that default will occur during the 3 year period?

  1. 0.1247%
  2. 0.1276%
  3. 0.1299%
  4. 0.1355%

The cumulative probability of default= 1-{Product of marginal probabilities of not defaulting
= 1-{(1-0.0003)*(1-.0004)*(1-0.0006)}
= 0.001299
Therefore the cumulative probability of default is 0.1299%


  • Assumptions:
    • The probability of observing a single event over a small interval is approximately proportional to the size of that interval
    • The probability of an event within a certain interval does not change over different intervals
    • The probability of an event in one interval is independent of the probability of an event in any other interval which is not overlapping
  • Poisson distribution is a special case of Binomial distribution when the probability of success (p) becomes very small and the number of events (n) becomes very large in such a way that the product of both gives a constant (λ).
    • Fix the expectation λ =n * p
    • Number of trials n
    • A Binomial distribution will become a Poisson distribution
  • E[X] = λ, Var(X) = λ



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