1. Fractiles Given a probability distribution F(x) and a number p, we define a p-fractile x p by the following formulas

2. For a continuous increasing F(x) a fractile is actually the inverse of F(x) For a distribution F(x) with jumps, we can determine fractiles as follows for any

3. F(x)F(x) x p xpxp In intervals where the distribution is constant, the corresponding fractile is not determined uniquely

4. Special fractiles x 0.01 - percentile: x 0.35 - 35-th percentile x 0.25 - first quartile x 0.50 - second quartile = median x 0.75 - third quartile

5. Let, for a discrete random variable, the probability distribution be given by the probability function xixi x1x1 x2x2 x3x3...xnxn pipi p1p1 p2p2 p3p3 pnpn If and we take

6. Example Calculate the median of a discrete random variable whose probability function is given by the following table xixi 1.21.51.82.12.42.73.03.3 pipi 0.120.250.180.010.150.20.040.05 We have 0.12 + 0.25 = 0.37 and 0.12 + 0.25 + 0.18 = 0.55 so that

7. Important distributions of discrete random variables  binomial distribution  Poisson distribution

8.  uniform distribution  normal distribution  chi-square distribution  t-distribution  F distribution Important distributions of continuous random variables

9. Binomial distribution A binomial experiment has four properties: 1)it consists of a sequence of n identical trials; 2)two outcomes, success or failure, are possible on each trial; 3)the probability of success on any trial, denoted p, does not change from trial to trial; 4)the trials are independent.

10. Let us conduct a binomial experiment with n trials, a probability p of success and let X be a random variable giving the number of successes in the binomial experiment. Then X is a discrete random variable with the range {0, 1, 2,..., n} and the probability function We say that X has a binomial distribution.

13. Expectancy and variance of a binomial distribution For a random variable X with a binomial distribution Bi(n,p) we have

14. Using the identity we can calculate The same identity can be used to calculate D(X)

16. Poisson distribution Poisson distribution is defined for a discrete random variable X with a range {0, 1, 2, 3,... } denoting the number of occurren- ces of an event A during a time interval if the following conditions are fulfilled: 1) given any two occurrences A 1 and A 2, we have 2) the probability of A occurring during an interval with equals to where c is fixed 3) denoting by P 12 the probability that E occurs at least twice between t 1 and t 2, we have

17. The Poisson distribution Po(λ) is given by the probability function

19. The expectancy and variance of a random variable X with Poisson distribution are given by the following formulas

21. Relationship between Bi(n,p) and Po(λ) For large n, we can write

23. Normal distribution law A continuous random variable X has a normal distribution law with E(X) = μ and D(X) = σ if its probability density is given by the formula

24. Standardized normal distribution A continuous random variable X has a standardized normal distribution if it has a normal distribution with E(X) = 0 and D(X) = 1 so that its probability density is given by the formula

25. Standardized normal distribution law (Gaussian curve)

26. There are several different ways of obtaining a random variable X with a normal distribution law. For large n, binomial and Poisson distributions asymptotically approximate normal distribution law If an activity is undertaken with the aim to achieve a value μ, but a large number of independent factors are influencing the performance of the activity, the random variable describing the outcome of the activity will have a normal distribution with the expectancy μ. Of all the random variables with a given expectancy μ and variance σ 2, a random variable with the normal distribution N(μ,σ 2 ) has the largest entropy.