# 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?

- 5%
- 50%
- 32%
- 3%

**C.**

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?

2.4%

2.5%

97.4%

97.5%

**C.**

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) = 0.9^{1/4}

= 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?

- 0.1247%
- 0.1276%
- 0.1299%
- 0.1355%

**C.**

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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