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Some Probabilistic Models Bernoulli Trials 1 Some probabilistic models Most common discrete random variables Bernoulli trials Binomial distribution Poisson distribution Most common continuous random variables Uniform distribution Exponential distribution Normal distribution Central Limit Theorem Other distributions related to the Normal (Chapter 4. Introduction to statistical inference) Statistics I. ECO/ Double courses ECO-DER and ADE-INF 2010/11 Chapter 1 2 Bernoulli trials Description / Definition It is a way to model statistically any experiment that has only two possible results , mutually exclusive, that they are called success and failure , with the condition that the probability of these two results does not change each time the experiment is carried out.(Bernoulli trials or experiments). If the probability of success is p (therefore, of failure is 1 p), the Bernoulli random variable is defined as − 1, if a success is observed , X = 0, if a failure is observed . Support of X : S = 0, 1 , with probabilities P(X = 0) = 1 p = q, P(X = 1) = p. { } − To denote that X follows a Bernoulli distribution with parameter p we write X Ber (p). ∼ Statistics I. ECO/ Double courses ECO-DER and ADE-INF 2010/11 Chapter 1 3 Bernoulli trials Example Result of tossing a coin 1, if heads , X = 0, if tails . It is Bernoulli trial, where as success we consider the heads. X follows a Bernoulli distribution with parameter 1 /2 (if the coin is fair). Example An airline has determined that the passengers that purchase tickets for a given flight have probability 0 .05 of not showing up at the airport We define 1 if the passenger shows up Y = 0 if he/she does not show up Y follows a Bernoulli distribution with parameter 0 .95 Statistics I. ECO/ Double courses ECO-DER and ADE-INF 2010/11 Chapter 1 4 Bernoulli trials Probability function: P(X = 0) = 1 p P (X = 1) = p − Distribution function: 8 < 0, if x < 0 − ≤ F (x) = : 1 p, if 0 x < 1 1, if x ≥ 1 Properties E(X ) = 0 P(X = 0) + 1 P(X = 1) = 0 (1 p) + 1 p = p − E(X 2) = 0 2 P(X = 0) + 1 2 P(X = 1) = 0 2 (1 p) + 1 2 p = p − V (X ) = E(X 2) E(X )2 = p p2 = p(1 p) = pq − − − Statistics I. ECO/ Double courses ECO-DER and ADE-INF 2010/11 Chapter 1 5 Binomial Distribution Description / Definition Consider independent Bernoulli trials that they are repeated n times with the same probability of success p. The r.v. X that counts the number of observed successes in the n trials is said to follow a Binomial distribution with parameters n and p and we write X B(n, p). ∼ The r.v. X takes values in S = 0, 1, 2,..., n and its probability function is given by the formula { } n x n−x P(X = x) = p (1 p) , x = 0 , 1,..., n, 0 p 1, x − ≤ ≤ n where = n! , for 0 x n. Recall that, for convenience, 0! = 1. x x!( n−x)! ≤ ≤ Properties E(X ) = np , V (X ) = np (1 p) = npq . − Statistics I. ECO/ Double courses ECO-DER and ADE-INF 2010/11 Chapter 1 6 Binomial Distribution Example The airline in the preceding example has sold 80 tickets for a given flight. The probability of a passenger not showing up is equal to 0 .05. Define X = number of passengers that show up . Then (assuming independence) X B(80 , 0.95) ∼ The probability that all 80 passengers show up is 80 − P(X = 80) = 0.95 80 (1 0.95)80 80 = 0 .0165 80 × − The probability that at least one passenger does not show up is P(X < 80) = 1 P(X = 80) = 1 0.0165 = 0 .9835 − − Statistics I. ECO/ Double courses ECO-DER and ADE-INF 2010/11 Chapter 1 7 Binomial Distribution: probability function The probability function of X B(80 , 0.95) is ∼ Binomial Distribution: Trials = 80, Probability of success = 0.95 Probability Mass 0.00 0.05 0.10 0.15 0.20 68 70 72 74 76 78 80 Number of Successes Changing the probability of success: Binomial Distribution: Trials = 80, Probability of success = 0.5 Binomial Distribution: Trials = 80, Probability of success = 0.1 Probability Mass Probability Mass 0.00 0.05 0.10 0.15 0.00 0.02 0.04 0.06 0.08 25 30 35 40 45 50 55 5 10 15 Number of Successes Number of Successes Statistics I. ECO/ Double courses ECO-DER and ADE-INF 2010/11 Chapter 1 8 Poisson distribution: counting rare events Description / Definition Counts the number of rare events that take place in a given period of time, or in a predefined space. For example, number of phone calls in an hour, number of typos in a page, number of traffic accidents in a week, . A random variable X follows a Poisson distribution with parameter λ, it is denoted X Pois (λ), if its probability function is ∼ x −λ λ P(X = x) = e , para x = 0 , 1, 2,... x! Observe that X takes values in S = 0, 1, 2,... = N 0 . { } ∪ { } Properties E(X ) = λ, V (X ) = λ. λ represents the mean number of events that occur per time or space unit. Exponencial Statistics I. ECO/ Double courses ECO-DER and ADE-INF 2010/11 Chapter 1 9 Poisson distribution: counting rare events Property of Poisson If X Pois (λ) and represents the number of rare events in time or space unit,∼ and Y is a r.v. that represents the number of these rare events in s units , then: Y Pois (sλ) ∼ Statistics I. ECO/ Double courses ECO-DER and ADE-INF 2010/11 Chapter 1 10 Poisson distribution: counting rare events Example The mean number of typos per slide is 0.2. Let X the r.v. that counts the number of typos per slide, then X Pois (0 .2) ∼ What is the probability that in one slide there are no typos? 0 − . 0.2 − . P(X = 0) = e 0 2 = e 0 2 = 0 .8187. 0! What is the probability that in 4 slides there is exactly one typos? Let Y the r.v. that counts the number of typos in 4 slides. Then: Y Pois (0 .2 4) = Pois (0 .8) ∼ 1 · − . 0.8 − . P(Y = 1) = e 0 8 = e 0 8 0.8 = 0 .3595. 1! Statistics I. ECO/ Double courses ECO-DER and ADE-INF 2010/11 Chapter 1 11 Uniform Distribution Description / Definition It is said that a variable X follows a uniform distribution in the interval (a, b), and it is denoted by X (a, b), if its density function is ∼ U 1 , if x (a, b), f (x) = b−a ∈ 0, if x / (a, b). ∈ This r.v. is defined by its minimum and maximum values, that is, a and b are its parameters. Properties a+b (b−a)2 E(X ) = 2 , V (X ) = 12 . Statistics I. ECO/ Double courses ECO-DER and ADE-INF 2010/11 Chapter 1 12 Uniform Distribution Statistics I. ECO/ Double courses ECO-DER and ADE-INF 2010/11 Chapter 1 13 Exponential distribution Description / Definition The exponential distribution models the elapsed time between two events that occur independently, separately and uniformly in time. It is said the the r.v. X follows a exponential distribution with parameter λ, and it is denoted by X exp (λ), if its density function is ∼ −λx f (x) = λ e , para x 0. ≥ Observe that X takes values in the set S = [0 , + ). ∞ Examples Time between the arrivals of two trucks at a discharge point. Time between emergency calls. Lifetime of a light bulb. Statistics I. ECO/ Double courses ECO-DER and ADE-INF 2010/11 Chapter 1 14 Exponential distribution Statistics I. ECO/ Double courses ECO-DER and ADE-INF 2010/11 Chapter 1 15 Exponential distribution Properties 1 E(X ) = λ 1 V (X ) = λ2 Distribution funcion: 1 e−λx , if x 0, F (x) = − ≥ 0, if x < 0. The exponential is related to the Poisson distribution. Poisson λ is the number of occurrences of an event per unit of time. Statistics I. ECO/ Double courses ECO-DER and ADE-INF 2010/11 Chapter 1 16 Exponential distribution Example On average, there are 50 serious house fires each year in a specific area. We know that the number of fires each year follows a Poisson distribution. What is the mean time between two fires? If a fire has just taken place, what is the probability that the next fire occurs after two weeks? We know that: The number of fires N (λ) with λ = 50 . ∼ P The time between two fires X (λ) with λ = 50 . ∼ E 1 The average time between two fires E[X ] = λ = 1 /50 years, 7.3 days. 2 Two weeks expressed as a fraction of years is: 52 = 0 .038 P[X > 0.038] = 1 P[X 0.038] = 1 (1 e−50 ·0.038 ) = 0 .15 − ≤ − − Statistics I. ECO/ Double courses ECO-DER and ADE-INF 2010/11 Chapter 1 17 Normal distribution Description / Definition The normal distribution describes an ideal r.v. It is a theoretical model that approximates very well real situations. The statistical inference is fundamentally based on the normal distribution and on distributions that are derived from the normal. It is said that a r.v. X follows a normal distribution or gaussian with parameters µ and σ, and it is denoted by X (µ, σ ), if its density function is ∼ N 1 1 f (x) = exp (x µ)2 σ √2 π −2 σ2 − Properties E(X ) = µ, V (X ) = σ2.
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