FN415.02 50
A Bernoulli trials are identical independent experiments
with only two possible outcomes (success or fail), e.g. tossing a coin, inspecting a product to see if it is
defective.
In the case of consecutive Bernoulli trials, each
elementary event can be described by a sequence like consisting of digits.
( success, fail)
Because of the independence of the trials, the
probability of an elementary event with successes and failures is
Note: the elementary events are equiprobable if and only if .
Some Important Probability Distributions
Let be the random number representing the total number of (success) in Bernoulli trials, i.e. if success occur in the
elementary event .
The number of distinct elementary events with the same total number of successes is , and each of these events has the same probability
So, the event has the probability
The probability distribution of this random variable is called the binomial distribution
Some Important Probability Distributions
FN415.02 52
Exercise: If we toss a fair coin five times, what is the
probability of obtaining 2 heads?
Some Important Probability Distributions
Exercise: If we toss a fair coin five times, what is the
probability of obtaining less than three heads?
The events of obtaining or heads are mutually
exclusive, so their probabilities add.
Some Important Probability Distributions
FN415.02 54
Exercise: Suppose we measure the performance of mutual funds by whether they outperform or underperform the
benchmark index. Assume that all the funds are independent, and have equal chances to be above or below the market in any given year. Let the total number of funds is .
What is the probability that at least one mutual fund will outperform the benchmark index for consecutive years? We are modeling the mutual funds as independent
sequences of Bernoulli trials. We also know that Bernoulli trials are distributed by binomial distribution.
Given that there are independent binomial random variables, what is the probability that at least one of them is ?
Some Important Probability Distributions
For each fund, the probability to have at least one below- benchmark year, is
The probability that all of them will have at least one below- benchmark year, i.e. , is
The probability that at least one of them will outperform consecutive years is
Suppose , we have
Even if the funds’ performance is random, it is almost
guaranteed that some of them will consistently outperform the benchmark for 5 years in a row!!
By the law, every fund prospectus must say that past
Some Important Probability Distributions
FN415.02 56
Exercise: Suppose a mortgage-backed security of
duration one year is based on 1-year underlying
mortgages. Assume that, for each mortgage, one of two things can happen:
1) The mortgage pays back interest at the end of the year.
2) The mortgage defaults with loss.
The probability of default is for each mortgage. Suppose that the MBS consists of two tranches: 1) The senior tranche which has of the capital. 2) The equity tranche which has of the capital.
The senior tranche earns interest, the equity tranche earns .
Some Important Probability Distributions
What is the probability that the senior tranche will lose some capital, if the defaults of individual mortgages are
independent?
The event that the senior tranche will lose some capital is the event that five or more out of the ten mortgages will default. If the defaults are independent, the probability that exactly mortgages out of will default is:
So, the probability of at least five mortgages default is or, alternatively,
Some Important Probability Distributions
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What is the probability that the senior tranche will
completely wiped out if the defaults of individual mortgages are perfectly dependent?
If the defaults happen altogether, the probability of at least five defaults is the same as the probability of a single default, which is .
Some Important Probability Distributions
What is the probability that the senior tranche is
completely wiped out if the defaults are independent? This happens when all ten mortgages default. So,
Some Important Probability Distributions
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A random variable taking only the integer is said to
have a Poisson distribution if
The distribution is specified by a single positive parameter , equal to the mean value of
It can be shown that the binomial distribution can be approximated by Poisson distribution with parameter (the average number of success), if is large and the probability of success is small.
Some Important Probability Distributions
Exercise: How many lottery tickets must be bought to make the probability of winning at least ?
Let be the total number of lottery tickets and the total number of winning tickets. Then each ticket has the
probability of winning . The purchase of each ticket can be regarded as an independent trial with the probability of success (small), where is the number of tickets bought. We usually need large . Then, must also be large (buy many tickets), and the number of winning tickets becomes Poisson random variable.
Some Important Probability Distributions
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Normal distributions are enormously important in
probability and statistics. This is due to the central limit theorem (will be discussed next session).
Note: important in theory does not implies important in
real financial world.
A normal distribution is specified by two parameters: a
mean and variance .
Skewness = 0 Kurtosis = 3