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Fractiles Given a probability distribution F(x) and a number p, we define a p-fractile xp by the following formulas lim F x p x x p lim F x p x x p For a continuous increasing F(x) a fractile is actually the inverse of F(x) F x p p x p F 1 p For a distribution F(x) with jumps, we can determine fractiles as follows F x p1 , F x p2 x p x for any p1 p p2 x F(x) p x xp In intervals where the distribution is constant, the corresponding fractile is not determined uniquely Special fractiles x0.01 - percentile: x0.35 - 35-th percentile x0.25 - first quartile x0.50 - second quartile = median x0.75 - third quartile Let, for a discrete random variable, the probability distribution be given by the probability function xi x1 x2 x3 ... xn pi p1 p2 p3 ... pn k If p i 1 i k 1 and p i 1 i we take f xk xk 1 xk k pi i 1 pk 1 Example Calculate the median of a discrete random variable whose probability function is given by the following table xi pi 1.2 1.5 1.8 2.1 2.4 2.7 3.0 3.3 0.12 0.25 0.18 0.01 0.15 0.2 0.04 0.05 We have 0.12 + 0.25 = 0.37 and 0.12 + 0.25 + 0.18 = 0.55 so that 0.5 0.37 x0.5 1.5 1.8 1.5 1.71667 0.18 Important distributions of discrete random variables binomial distribution Poisson distribution Important distributions of continuous random variables uniform distribution normal distribution chi-square distribution t-distribution F distribution 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. 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 pi Cni p i 1 p n i We say that X has a binomial distribution. Binomial distribution with n=10, p=0.5 0,3 0,25 0,2 0,15 p(x) 0,1 0,05 0 0 2 4 6 8 10 12 Binomial distribution with n=10, p=0.25 0,3 0,25 0,2 0,15 p(x) 0,1 0,05 0 0 2 4 6 8 10 12 Expectancy and variance of a binomial distribution For a random variable X with a binomial distribution Bi(n,p) we have E X pn D X np1 p Using the identity iCni i n! n n 1! nCni 11 i !n i ! i 1!n i ! we can calculate n E X iC p q i n i 0 i n i n iC p q i 1 i n i n i n n 1 i 1 i 0 n nCni 11 p i q n i i 1 p nCni 11 p i 1q n i pn Cni 1 p i q n i pn p q The same identity can be used to calculate D(X) n 1 pn n n i 0 i 1 E X 2 i 2Cni p i q n i np iCni 11 p i 1q n i n i 1 i 1 n i n np Cn 1 p q i 1Cni 11 p i 1q n i i 1 i 1 n n 1 i i n i i 2 i 2 n i np Cn 1 p q n 1 p Cn 2 p q i 0 i 2 n2 n 1 np p q n 1 p Cni 2 p i q n i 2 i 0 np 1 n 1 p p q n2 np1 n 1 p np1 p np npq np 2 D X E X 2 E X D X npq 2 Poisson distribution Poisson distribution is defined for a discrete random variable X with a range {0, 1, 2, 3, ... } denoting the number of occurrences of an event A during a time interval T1 ,T2 if the following conditions are fulfilled: 1) given any two occurrences A1 and A2, we have P A2 | A1 P A2 2) the probability of A occurring during an interval t1 ,t2 with T1 t1 t2 T2 equals to ct2 t1 where c is fixed 3) denoting by P12 the probability that E occurs at least twice between t1 and t2, we have P12 0 t1 t 2 t t 2 1 lim The Poisson distribution Po(λ) is given by the probability function p x e e x 0 x x x! 2 3 e 1 e e 1 x! 1! 2! 3! Poisson law with E(X) = 5 0,2 0,15 0,1 p(x) 0,05 0 0 5 10 15 The expectancy and variance of a random variable X with Poisson distribution are given by the following formulas E X D X E X e x x! x 0 E X 2 e x x! x 0 e x 0 x e x x 1 x 2 e x 1 x! x 1 x e x x! x e x 1 x 1! x 0 x! x 2 e x 1 x 1 x 1! x x x x x 1 e x e e e x! x 0 x! x 0 x! 2 D X E X 2 E X 2 2 2 Relationship between Bi(n,p) and Po(λ) For large n, we can write Bi(n, p) Po(np) Comparing Bi(20,0.25) with Po(5) Poisson 18 15 12 9 6 Binomial 3 0 0,25 0,2 0,15 0,1 0,05 0 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 1 f x e 2 x 2 2 2 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 1 f x e 2 x2 2 Standardized normal distribution law (Gaussian curve) 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 σ2, a random variable with the normal distribution N(μ,σ2) has the largest entropy.