¡Descarga TOPIC 1: MAIN PROBABILITY DISTRIBUTIONS y más Apuntes en PDF de Estadística Empresarial solo en Docsity!
TOPIC 1: MAIN PROBABILITY DISTRIBUTIONS
A) DISCRETE DISTRIBUTIONS
A.1. BINOMIAL
a) Binomial (1;p):
Characteristics:
- A random phenomenon occurring just once
- Two possible outcomes (dichotomy): A (yes) y Ā (no)
- Incompatible events
- One of those events will take place necessarily
- Formulation: 1 ( 1) 0 ( 0) P p P q
p q 1
Probability distribution: 1
x x P x p q x
Cumulative distribution function: F x ( ) P( x) 0 ; ( ) 0 0 1; ( ) 1 ( ) 1 x F x x F x q x F x
Expected value: E^ (^ ^ ) p Variance: V^ ( )^ ^ ^ p q Standard deviation: ^ ^ p q
b) Binomial (n;p):
Characteristics:
- Random phenomenon based on the realization “n” times of a dichotomous phenomenon.
- The trials are independent, meaning that an outcome does not affect subsequent outcomes
- Interpretation: the number “x” of successes obtained from n possible dichotomic outcomes Formulation of the DRV:
ξ = ξ 1 + ξ 2 + … + ξi+ … + ξn
where ξ ~ B(1;p) independent Then ξ ~ B(n;p) Probability distribution: ௫ ି௫ 0 ( ) 1 n x P x ^ ^
Approximation rule: Another perspective: this distribution is useful to determine probabilities for random events occurring in continuous fixed intervals (of time and space) o Those random events are dichotomic o The process is stable meaning that, on the long term, an average number of events per unit of time or space occur o Occurrences are independent. Therefore, the number of occurrences in a given unit is independent of the number of occurrences in any other nonoverlapping unit o In this context, two situations can be considered: The time elapsed between the occurrence of two consecutive events (exponential distribution) The number of events happening in an interval of time (Poisson’s distribution)
- Examples: The number of breakdowns, during a month, in certain machine The number of clients entering a branch from 10 am to 11 am The number of faulty articles in a shipment of 1000 units The number of claims in a life insurance company for a given day The number of phone calls received, per second, in an office The number of landings in an airport, every 15 minutes
- It is also called “the law of rare events” because the interval of time, in which the occurrence of the events is studied, can be divided in small subintervals where the happening of such event is B(1;p) with p → 0 Probability distribution:
x
P x e x
x
Cumulative distribution function: F x( )^ ^ P(^ x) 0 0 ; ( ) 0 0 1; ( ) ( 0) 1 2; ( ) ( 0) ( 1) 2 3; ( ) ( 0) ( 1) ( 2) ............... ( ) ( ) 1 n x x F x x F x P x F x P P x F x P P P x n F x P x
(^) Expected value and variance: E ( ) V( ) Standard deviation:^ ^ Additive property: Let ξ 1 ~ P(λ 1 ) and ξ 2 ~ P(λ 2 ) independent
If γ = ξ 1 + ξ 2 then γ ~ P(λ 1 +λ 2 )
Variance: 2 2 2 2 3 3 2
b a
b a
b a
Standard deviation: 2 ( ) 12 b a
B.2. Normal distribution
o It is the most important distribution in Statistics o There is certain controversy in relation to the authorship of the discovery o Some authors consider it was discovered by De Moivre in 1773 as an approximation to the B(n;p) o But most concede this acknowledgement to Gauss, provided he was the first scientist in using the normal law to measure errors in experiments (1809) o Laplace was also a key author, given that he presented among other things the central limit theorem (1812) o The normal distribution approximates the probability distribution of many random variables, such as the B(n;p) and the Poisson o Central Limit Theorem: if a rp is the result of a high number of independent random phenomenon, each of them quantitative small, then its probability distribution approximates a normal distribution
o Probability distributions of sample means approach a normal distribution when the sample size is high o It is used in all the sciences requiring empirical experiments such as medicine, phisics, biology, economy, and so on. o Some distributions arising from the normal distribuiton are Pearson’s χ 2 , Student’s t and Snedecor’s F o We can distinguish the normal distribution and the standard normal distribution, the latter being a particular case of the former o We will start with the standard normal distribution and follow with the general case
B.2.a) Standard Normal distribution (0;1) or Z distribution:
o It is a normal distribution whose expected value equals cero and its standard deviation is one o It has a bell-shaped density function whose formula is: 2 x^2
e
f (x )
o Most of the values fall around the mean o It is simmetric about the mean o Half of the values are below cero and half above cero o Mean, median and mode are the same
o Probabilities are calculated by tables
f(x) - + o The probability in [μ-σ;μ+σ] is about 68% o The probability in [μ-2σ;μ+2σ] is about 95% o The probability in [μ-3σ;μ+3σ] is almost 100% o Any probability can be obtained by transforming the N(μ;σ) into the Z standard normal. This operation is called standardization: *^ ^
- Additive property: Let ξi ~ N(μi;σi) independent; i = 1..n a, b 1 , b 2 , …, bn constants and w = a + b 1 ξ 1 + b 2 ξ 2 +…+ bnξn Then: 𝑤~𝑁 ൮𝑎 + 𝑏𝜇 ; ඩ 𝑏 ଶ 𝜎 ଶ ୀଵ ୀଵ ൲
B.3. Pearson’s Chi-square distribution:
ଶ o It was developed by K. Pearson at the beginning of the 20th century o There is not any event in the reality following this distribution Definition: Then: ଶ ଵ ଶ ଶ ଶ ଷ ଶ ଶ ଶ ୀଵ
- It only takes positive values
- Expected value and variance:
E x n
V x n
1 g.l. 4 g.l. 10 g.l.
B.4. t Student distribution:
- It was obtained by W.S. Gosset in 1908
- It does not correspond to any situation coming from the real life Definition: Then: ଵ ଶ ଶ ଶ ଷ ଶ ଶ ଶ Expected value and variance (being n > 2): ( ) 0 ( ) 2 E x n V x n
- Its density function is simmetric and bell shaped
- It presents more dispersión than Z, but it lowers when n increases. 1 g.l. 4 g.l. 9 g.l.
- It will be appropriate in estimating population means from normal variables with unknown variances
- It approaches Z when n increases:
B.5. Fisher-Snedecor F distribution: ,
Definition: Then: , ଵ ଶ ଶ ଶ ଷ ଶ ଶ ଵ ଶ ଶ ଶ ଷ ଶ ଶ ଶ ଶ
