STAT 430/510 Probability Lecture 12: Central Limit Theorem and Exponential Distribution Pengyuan (Penelope) Wang June 15, 2011
Review Discussed Uniform Distribution and Normal Distribution Normal Approximation to Binomial Distribution
A little thing about Normal Distribution For N(µ, SD = σ), 68% probability is within ±σ around µ. 95% probability is within ±2σ around µ. 99.7% probability is within ±3σ around µ. So if a value is outside 3σ around µ, it is very rare to happen!
Central Limit Theorem: a Generalization of Normal Approximation to Binomial Distribution X i, i = 1,..., n independently and identically follow a distribution with expected value µ and standard deviation σ. If n is large, then X = 1 n (X 1 + X 2 +... + X n ) follows approximately a normal distribution with mean µ, and standard deviation σ n.
Example There are 100 workers in a factory. Assume that their ages independently and identically follows a distribution with expected value 40 years old and standard deviation 10 years old. What is the probability that their mean age higher than 45 years old? Let X represent their mean age. According to the Central Limit Theorem, X approximately follows a normal distribution with mean 40 and standard deviation 10/ 100 = 1. P(X > 45) = P(Z > 45 40 1 ) = P(Z > 5) = 0.00.
Example There are 100 male workers and 100 female workers in a factory. Assume that the ages of males independently and identically follows a distribution with expected value 40 years old and standard deviation 10 years old. Assume that the ages of females independently and identically follows a distribution with expected value 39 years old and standard deviation 20 years old. What is the probability that male s mean age is at least 2 years higher than female s mean age?
Example There are 100 male workers and 100 female workers in a factory. Assume that the ages of males independently and identically follows a distribution with expected value 40 years old and standard deviation 10 years old. Assume that the ages of females independently and identically follows a distribution with expected value 39 years old and standard deviation 20 years old. What is the probability that male s mean age is at least 2 years higher than female s mean age? The male s mean age (X) approximately follows normal distribution (40, 1), and the female s mean age (Y ) follows normal distribution (39, 2). Thus X Y approximately follows normal distribution (1, 5). So P(X Y > 2) P(Z > (2 1)/ 5) = 1 Φ(1/ 5).
Example The approximation to the Binomial distribution is just one application of the central limit theorem. An insurance company believes that people can be divided into two classes: those who are accident prone and those who are not. The company s statistics show that an accident-prone person will have an accident at some time within a fixed 1-year period with probability 0.4, whereas this probability decreases to 0.2 for a person who is not accident prone. We assume that 30 percent of the population is accident prone. If we draw 100 policyholders, what is the probability that more than 30 of them have accidents within a year of purchasing a policy?
Example The probability to have accident for each person is 0.4*0.3+0.2*0.7=0.26. Let X be the proportion of the people with accidents, then X 0.26 0.26 0.74/100 approximately follows standard Normal distribution. X 0.26 P(X > 0.3) = P( > 0.26 0.74/100 1 Φ(0.91). 0.3 0.26 ) = 0.26 0.74/100
Exponential Random Variable A continuous random variable X is said to have a exponential distribution with parameter λ if the pdf of X is { λe f (x) = λx, if x 0 0, if x < 0
cdf of Exponential r.v. For exponential r.v. X with parameter λ, the cdf is { 1 e F(x) = λx, x 0 0, x < 0
Expected Value and Variance X is exponential random variable with parameter λ. E[X] = 1 λ Var(X) = 1 λ 2
Example Suppose that the length of a phone call in minutes is an exponential random variable with parameter λ = 0.1. If someone arrives immediately ahead of you at a public telephone booth, find the probability that you will have to wait (a) More than 10 minutes? (b) Between 10 and 20 minutes?
Example: Solution Let X denote the length of the call made by the person in the booth. P(X > 10) = 1 F(10) = e 1 = 0.368 P(10 < X < 20) = F(20) F(10) = e 1 e 2 = 0.233
Memoryless Property The exponential distribution is memoryless! It means: if for all s, t 0. P(T s + t T t) = P(T s) In fact, the exponential distribution is the only memoryless continuous distribution.
Example Suppose that the length of a phone call in minutes is an exponential random variable with parameter λ = 0.1. Someone arrives immediately ahead of you at a public telephone booth. Now you have already waited for 10 minutes, from now on what is the probability that you need to wait at least for another 10 minutes?
Example Suppose that the length of a phone call in minutes is an exponential random variable with parameter λ = 0.1. Someone arrives immediately ahead of you at a public telephone booth. Now you have already waited for 10 minutes, from now on what is the probability that you need to wait at least for another 10 minutes? From the memoryless property of exponential distribution, P(X > 10 + 10 X > 10) = P(X > 10) = e 10 0.1 = e 1
Example Suppose that the number of miles that a car can run before its battery wears out is exponentially distributed with an average value of 10,000 miles. If a person has used the battery for some time and now he desires to take another 5000-mile trip, then what is the probability that he or she will be able to complete the trip without having to replace the car battery? What can be said when the distribution is not exponential?
Example: Solution X is the lifetime of the battery t is the number of miles that the battery had been in use prior to the start of the trip. From the memoryless property of exponential distribution, P(X > t+5000 X > t) = P(X > 5000) = e 5000/10000 = 0.606 If the distribution F of X is not exponential distribution, P(X > t+5000 X > t) = P(X > t + 5000) P(X > t) = 1 F(t + 5000) 1 F(t)