Some standard univariate probability distributions

1. Some standard univariate probability distributions • Characteristic, moment generating, cumulant generating functions • Discrete distribution • Continuous distributions • Some distributions associated with normal • References

2. Characteristic, moment generating and cumulant generating functions Characteristic function is defined as the expectation of the function - eitx Moment generating function is defined as (the expectation of etx ): Moments can be calculated in the following way. Obtain derivative of M(t) and take the value of it at t=0 Cumulant generating function is defined as the natural logarithm of the characteristic function

3. Discrete distributions: Binomial Let us assume that we carry out an experiment and result of the experiment is either “success” or “failure”. Probability of “success” is p. Then probability of failure will be q=1-p. We carry experiments n times. What is the probability of k successes: Characteristic function: Moment generating function: Find first and second moments

4. Discrete distributions: Poisson When the number of trials (n) is large and the probability of successes (p) is small and np is finite and tends to  then binomial distribution converges to Poisson distribution: Poisson distribution describes the distribution an event that occurs rarely in a short period. It is used for example in counting statistics to describe the number of the registered photons. Characteristic function is: What is the first moment?

5. Discrete distributions: Negative Binomial Consider experiment: Probability of “success” is p and probability of “failure” is q=1-p. We carry out experiment until k-th success. We want to find probability of j failures. (It is called sequential sampling. Sampling is carried out until some stopping rule is satisfied). If we have j failure then it means that the number of trials is k+j. Last trial was success. Then probability that we will have exactly j failures is: It is called negative binomial because coefficients are from negative binomial series: p-k=(1-q)-k Characteristic function is: What is the moment generating function? What is the first moment?

6. Continuous distributions: uniform Simplest form of the continuous distribution is the uniform with density: Distribution is: Moments and other properties are calculated easily.

7. Continuous distributions: exponential Density of exponential distribution has the form: This distribution has two origins. • Maximum entropy. If we know that random variable is non-negative and we know its first moment – 1/ then maximum entropy distribution has the exponential form. • From Poisson type random processes. If probability distribution of j(x) events occurring during time interval (0;x] is a Poisson with mean value x then probability of time elapsing till the first event occurs has the exponential distribution. Let Tr denotes time elapsed untilr-th event Putting r=1 we get e(- x). Taking into account that P(T1>x) = 1-F1(x) and getting its derivative wrt t we arrive to exponential distribution This distribution together with Poisson is widely used in reliability studies, life testing etc.

8. Continuous distributions: Gamma Gamma distribution can be considered as a generalisation of the exponential distribution. It has the form: It is the probability of time x elapsing before r events happens Characteristic function of this distribution is: This distribution is widely used in many application. One of the application is the use in prior probability generation for sample variance. For this inverse Gamma distribution is used (by changing variable y = 1/x we get inverse Gamma). Gamma distribution can be generalised to non-integer values of r also (by putting (r) instead of (r-1)! )

9. Continuous distributions: Normal Perhaps the most popular and widely used continuous distribution is the normal distribution. Main reason for this is that usually random variable is the sum of the many random variables. According to central limit theorem under some conditions (for example: random variables are independent. first and second moments exist and finite then distribution of sum of random variables converges to normal distribution) Density of the normal distribution has the form Another remarkable fact is that if we know only know mean and variance then maximum entropy distribution is the normal. Its characteristic function is:

10. Exponential family Exponential family of distributions has the form Many distributions are special case of this family. Natural exponential family of distributions is the subclass of this family: Where A() is a natural parameter. If we use the fact that distribution should be normalised to 1 then characteristic function of the natural exponential family with natural parameter A() =  can be derived to be: Try to derive it. Hint: use the normalisation fact. Find D() and then use expression of characteristic function and D() . This distribution is used for fitting generalised linear models.

11. Continuous distributions: 2 Normal variables are called standardized if their mean is 0 and variance is 1. Distribution of the sum of the squares of n standardized normal random variables is 2 with n degrees of freedom. Density function is: If there are p linear restraints on the random variables then degree of freedom becomes n-p. Characteristic function for this distribution is: 2is used widely in statistics for such tests as goodness of fit of a model to the experiment.

12. Continuous distributions: t and F-distributions Two more distributions are closely related with the normal distribution. One of them is Student’s t-distribution. It is used to test if mean value of the sample is significantly different from 0. Another and similar application is for tests of differences of means of two different samples. Distribution of ratio of the random variables with standardised normal distribution to the square root of the random variable with 2 distribution is t-distribution. Fisher’s F-distribution is a distribution of ratio of the variances of two different samples. It is used to test if their variances of two samples are different. One of the important application is in ANOVA.

13. Reference Johnson, N.L. & Kotz, S. (1969, 1970, 1972) Distributions in Statistics, I: Discrete distributions; II, III: Continuous univariate distributions, IV: Continuous multivariate distributions. Houghton Mufflin, New York. Mardia, K.V. & Jupp, P.E. (2000) Directional Statistics, John Wiley & Sons.