Chapter 2 Flashcards
Discrete Distributions
The distributions considered here are all models for the number of something, such as the number of successes, trials, deaths, or claims.
The values assumed by these variables are integers from the set {0, 1, 2, 3, …}. Such variables are often referred to as counting variables.
Another word for a Bernoulli variable
Indicator variable
Binomial distribution
A series of n Bernoulli trials where
the trials are independent of one another, ie the outcome of any trial does not depend on the outcomes of any other trials
the trials are identical
OR n IID Benroulli’s
Geometric distribution
Consider again a sequence of independent, identical Bernoulli trials with P({s})=p .
Thevariable of interest now is the number of trials that has to be performed until the first success occurs. Because trials are performed one after the other and a success is awaited, this distribution is one of a class of distributions called waiting-time distributions.
Type I Negative Binomial with k=1
OR
Another formulation of the geometric distribution is sometimes used. Let Y be the number of failures before the first success. Then
P(Y=y)=p(1-p)^y
and Y=X-1 (Type 2 Neg Binom)
The memoryless property
For the Geom, Exp distribution:
P(X>x+n|X>n)=P(X>x)
The lack of success on the first n trials is irrelevant – under this model the chances of success are not any better because there has been a run of bad luck
Negative Binomial distribution
Type I
The random variable X is the number of the trial on which the k th success occurs, where k is a positive integer.
Note: The mean and variance are just k times those for the geometric(p) variable, which is itself a special case of this random variable (with k=1 ). Further, the negative binomial
variable can be expressed as the sum of k geometric variables (the number of trials to the first success, plus the number of additional trials to the second success, plus … to the (k-1 )th success, plus the number of additional trials to the k th success.)
Type II
Let Y be the number of failures before the k th success.
Then
Y=X-k
DEFAULT FOR R
HyperGeometric
This is the ‘finite population’ equivalent of the binomial distribution.
Suppose objects are selected at random, one after another, without replacement, from a finite population consisting of k ‘successes’ and N-k ‘failures’. The trials are not independent, since the result of one trial affects the make-up of the population from which the next selection is made.
Random variable X : is the number of ‘successes’ in a sample of size n from a population of size N that has k ‘successes’ and N-k ‘failures’ .
P(X=x)=(kCx)(N-kCn-x)/(NCn), max{0,n-N+k<=x<=min{k,n}
E(X)=nk/N
Var(X)=nk(N-k)(N-n)/N^2(N-1)
Poisson distribution
This distribution models the number of events that occur in a specified interval of time, when the events occur one after another in time in a well-defined manner. This manner presumes that the events occur singly, at a constant rate, and that the numbers of events that occur in separate (ie non-overlapping) time intervals are independent of one another.
These conditions can be described loosely by saying that the events occur ‘randomly, at a rate of .. per ..’, and such events are said to occur according to a Poisson process.
Another approach to the Poisson distribution uses arguments which appear at first sight to be unrelated to the above. Consider a sequence of Binomial(n,p) distributions as n–>Infinity and p–>0 together, such that the mean np is held constant at the value λ. The limit leads to the distribution of the Poisson variable, with parameter λ.
Poisson recurrence relationship
P(X=x)=λ/x P(x=x-1)
Poisson approximation to the binomial
- Very good when n is large an dp is small.
- Typically n>=100, p<=0.05
- the value of np is all that matters (n=200,p=0.02 <=> n=400, p=0.01)
Gamma distribution
The parameter α changes the shape of the graph of the PDF, and the parameter λ changes the x scale
Exponential distribution
Gammma with α=1
The exponential distribution is used as a simple model for the lifetimes of certain types of equipment. Very importantly, it also gives the distribution of the waiting time, T, from one event to the next in a Poisson process with rate λ.
A gamma variable with parameters α=k (a positive integer) and λ can be expressed as the sum of k exponential variables, each with parameter λ. This gamma distribution is in fact the model for the time from any specified starting point to the occurrence of the k th event in a Poisson process with rate λ.
Beta distribution
- The (continuous) uniform distribution on (0,1) is a special case (with a=0 and b=1 ).
- The beta distribution is a useful distribution because it can be rescaled and shifted to create a wide range of shapes from straight lines to curves, and from symmetrical distributions to skewed distributions.
- Since the random variable can only take values between 0 and 1, it is often used to model proportions, such as the proportion of a batch that is defective or the percentage of claims that are over £1,000.
Normal distribution
This distribution, with its symmetrical ‘bell-shaped’ density curve is of fundamental importance in both statistical theory and practice.
Its roles include the following:
1. it is a good model for the distribution of measurements that occur in practice in a wide variety of different situations, for example heights, weights, IQ scores or exam scores.
2. it provides good approximations to various other distributions – in particular it is a limiting form of the binomial (n p,)
np .
3. It is also used to approximate the Poisson distribution.
4. it provides a model for the sampling distributions of various statistics
5. much of large sample statistical inference is based on it, and some procedures require an assumption that a variable is normally distributed.
6. it is a ‘building block’ for many other distributions.
A linear function of a normal variable is also a normal variable, ie if X is normally distributed, so is Y=aX+b .
Standard normal distribution
- It is not possible to find an explicit expression for Fx so X-mu/sd is used. Tables are provided for the distribution of Z
- it has mean 0 and standard deviation 1.
The distribution is symmetrical about 0. - The z-value is the number of standard deviations away from the mean it is
*if X has the standard normal distribution, then X^2 has the
chi-squared distribution with one degree of freedom (the special case of the gamma distribution).
lognormal distribution
- If X represents, for example, claim size and Y=logX has a normal distribution then X has a logNormal distribution
- The lognormal distribution is positively skewed and is therefore a good model for the distribution of claim sizes.
t distribution
If the variable X has a chi squared v distribution and the another independent variable Z has the
standard normal distribution, then
z/sqrt(X/v) =t dist with v degrees of freedom
F distribution
X, Y have chi squared dists with m, n degress of freedom then
F= X/m / Y/n
Poisson process
The Poisson process is an example of a counting process. Here the number of events occurring is of interest. Since the number of events is being counted over time, the number of events that take place in the time interval (0,t] ,must satisfy the following conditions
Counting process
1. N(0)=0 , ie no events have occurred at time 0.
2. For any t>0 , N(t) must be integer valued. ie we can’t have 2.3 claims.
3. When s< t , N(s)<=N(t) , ie the number of events over time is non-decreasing.
ie if we have counted, say, 5 deaths in 2 months, then the number of deaths counted in a 3 month period which includes the 2 month period must be at least 5.
4. When s< t , N(t)-N(s) represents the number of events occuring in the time interval (s,t)
Poisson Specifically
1. (1) and (3) from counting
2. P(N(t+h)=r|N(t)=r)=1-λh+o(h)
P(N(t+h)=r+1|N(t)=r)=λh+o(h)
P(N(t+h)>r+1|N(t)=r)=o(h)
where f(h)=o(h) if f(h)/h–>0 as h–>0
3. When s< t, the number of events in the time interval (s,t] is independent of the number of events up to time s
In other words, the numbers of events that occur in separate (ie non-overlapping) time intervals are independent of one another.
Condition (5) states that in a very short time interval of length h , the only possible numbers of events are zero or one. Condition (5) also implies that the number of events in a time interval of length h does not depend on when that time interval starts.
The reason why a process satisfying conditions (5) to (7) is called a Poisson process is that for a fixed value of t, the random variable N(t) has a Poisson distribution with parameter λt .
The waiting time between events is Exp(λ)
Inverse transform method for continuous distributions
- First, generate a random number u from the uniform U(0,1) distribution.
- To simulate a random variate X with probability density function (PDF) fX(x), we use the cumulative distribution function (CDF) FX(x).
- Let u be the probability that X takes on a value less than or equal to x, i.e.,
u = P(X ≤ x) = FX(x) - Solving for x, we get:
x = FX⁻¹(u) - Algorithm to generate a random variate x from a continuous distribution with CDF FX(x):
- Generate a random number u from U(0,1).
Return x = FX⁻¹(u).
The main disadvantage of the inverse transform method is the necessity to have an explicit expression for the inverse of the distribution function
F(x) .
