Statistics for Cybersecurity

Random number generators are important in many fields, including physics, engineering, mathematics, computer science and cryptographic applications (where they play a crucial role in some situations). Random number generators can be divided in two main classes: true random number generators (TRNG), which typically make use of an unpredictable physical mean to generate numbers (e.g. atmospheric noises);  and pseudorandom number generators (PRNG), which uses a mathematical deterministic algorithm, computed on an initial value (aka the seed) to generate the random numbers. Even though it should be obvious, is worth to insist on the fact that the choice of the random number generator strictly depend on what the generated numbers will be used for. For instance, not every PRG is good for cryptographic applications. Such a PRG should pass all the statistical tests another required condition is that an adversary, without knowing the seed, cannot distinguish its output from a truly ...

Law of Large Numbers and CLT Intuitively, everyone can be convinced of the fact that the average of many measurements of the same unknown quantity tends to give a better estimate than a single measurement. The law of the large numbers (LLN) and the central limit theorem (CLT) formalise this general ideas through mathematics and random variables. Suppose X 1 , X 2 , ..., X n are independent random variables with the same underlying distribution. In this case, we say that the X i are independent and identically-distributed (or, i.i.d.). In particular, the X i have all the same mean μ and standard deviation σ. The average of the i.i.d. variables is defined as: The central limit theorem states that when an infinite number of successive random samples are taken from a population, the sampling distribution of the means of those samples will become approximately normally distributed with mean μ and standard deviation σ/ √N as the sample size becomes larger, irrespective of the sh...

Chebyshev's Inequality In probability theory, the Chebyshev's inequality guarantees that, for a wide class of probability distributions, no more than a certain fraction of values can be more distant than a certain value from the mean. In particular, the mentioned inequality states that no more than 1/k 2 of the distribution's values can be more than k standard deviations away from the mean. In other words, this mean that at least (1 - 1/k 2 ) of the distribution's values are within k standard deviations of the mean. The Chebyshev's inequality can be easily derived from the Markov's inequality, where the latter defines an upper bound for the probability that a non-negative random variable is greater than (or equal to) some positive integer constant. Remember the Markov's inequality where   a  > 0 and  X  is a nonnegative random variable The Chebyshev inequality follows by considering the random variable ( X - E ( X )) 2  and the constant a 2...

Boole's inequality, which is also known as the union bound , states that for any finite  or countable  set of events, the probability that at least one of the events happens is no greater than the sum of the probabilities of the individual events. More formally, given a set of events A 1 , A 2 , A 3 , ..., A i the following inequality holds. The mentioned inequality may be proved in several ways, here the proof using induction is provided. Proof For the case n  = 1, we trivially have P ( A i ) ≤ P ( A i ). For the case n , we've to conclude with the inequality statement. Since  P ( A  ∪ B ) = P ( A ) + P ( B ) - P ( A  ∩ B ) and by the associative property of the union operator, we've: By the first axiom of probability, we've that the last term of the equality is greater than (or equal to) 0 and then Which is true for any A i , thus completing the proof. The discussed inequality may be generalised to find both the upper and the lo...

The are several kinds of mean, each with its applications and properties. In probability and statistics, the most used kind of mean is the arithmetic mean which is defined as the central value of a discrete set of numbers and can be computed by adding all the values and then dividing the result by the total number of values. For instance, the average age is equal to the sum of the ages of every individual in a particular sample, divided by the total number of individuals. Naturally, the sample mean will differ from the population mean but, however, by the law of large numbers with a larger sample size, its mean will be closer to the population mean. The following list summarises the other kind of means which are commonly used in statistics and mathematics. Weighted arithmetic mean is similar to an ordinary arithmetic mean except that, instead of each of the data points contributing equally to the final average, some data points contribute more than others. Geometric m...

Before giving the notion of conditional frequency first is important to define what is a contingency table, sometimes called two-way frequency table. A contingency table is a table which displays the (multivariate) frequency distribution of the variables of interest. For instance, consider the following dataset which contains information regarding a basket experiment made on a sample of 33 students. Boys Girls Totals Made 7 (21.2%) 6 (18.2%) 13 (39,4%) Missed 11 (33,3%) 9 (27.3%) 20 (60.6%) Totals 18 (54.5%) 15 (45.5%) 100% Every cell contains a joint relative frequency, which is the ratio of the frequency in a particular category and the total number of data values. As an example, 21.2% is obtained dividing 7 (the number of boys that made a basket) by the total of students, which is 33. Thus 7/33 = 21.2% is the joint relative frequency of boys that made a basket. The numbers in...

Statistics is a branch of mathematics which deals with the analysis, interpretation and presentation of data; it's used in several disciplines such as psychology, business, government, computer science and many more. In applying statistics it's convenient to begin with a statistical population to be studied. A statistical population is a set of elements which share some information and are of interest for one or more questions (or experiments). A subset of the population is called statistical  sample  and, a dataset , is a representation of the data that are considered during a statistical investigation. A dataset can be viewed as a table where every column is labeled with a variable name and each row contains the value the variable assume in a particular statistical unit. For instance, consider as the population the students of La Sapienza university and that we would like to analyse their gender, ages and their height. Below is the dataset of the information gathered a...