Sessions 1-20 15MT2005

# Session 11 uniform and exponential distribution

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Session: 11 Uniform and Exponential distribution: Uniform distribution: One of the simplest continuous distributions in all of statistics is the continuous uniform distribution. This distribution is characterised by a density function that is “flat” and thus the probability is uniform in a closed interval, say [A, B].

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102 Definition: The density function of the continuous uniform random variable X on the interval [A, B] is f ( x; A ,B ) = { 1 B A , A≤ x≤B 0 elsewhere Note: 1. The density function forms a rectangle with base B-A and a constant height 1/ (B-A). As a result, the uniform distribution is often called the rectangular distribution . The density function for a uniform random variable on the interval [1, 3] is shown in the following figure. 2. The mean and variance of the uniform distribution are μ = ( A + B ) 2 σ 2 = ( B A ) 2 12 3. The probability is uniform in the interval [0,1] is said to be a standard uniform distribution. Example: A bus arrives every 20 minutes at a specified stop beginning at 6:40 A. M. And continuing until 8:40 A. M. A certain passenger does not know the schedule, but arrives randomly (uniformly distributed) between 7:00 A. M. And 7:30 A. M. Every morning. What is the probability that the passenger waits more than 5 minutes for a bus? Solution: The passenger has to wait more than 5 minutes only if the arrival time is between 7:00 A. M. And 7: 15 A. M. Or between 7:20 A. M. And 7:30 A.M. If X is a random variable that denotes the number of minutes past 7:00 A. M. That the passenger arrives, the desired probability is P(0<X<15)+P(20<X<30) Now, X is a uniform random variable on (0,30). Therefore, the desired probability is given by F(15)+F(30)-F(20)=(15/30)+1-(20/30)=5/6. Problems to be discussed by the faculty:
103 1. Suppose that a large conference room for a certain company can be reserved for no more than 4 hours. However, the use of the conference room is such that both long and short conferences occur quite often. In fact, it can be assumed that length X of a conferences has a uniform distribution on the interval [0, 4]. a) What is the probability density function? b) What is the probability that any given conference lasts at least 3 hours? Exponential distribution: Exponential distribution plays an important role in both queuing theory and reliability. Time between arrivals at service facilities, and time to failure of components and electrical systems, often is nicely modelled by the exponential distribution. Definition: The continuous random variable X has an Exponential distribution, with parameter β (β>0), if its density function is given by f ( x ,β ) = { β e βx x > 0 0 elsewhere

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104 Note: 1) The mean and variance of the Exponential distribution are . 1 1 2 2 and 2) Cumulative distribution function F(x) of the Exponential distribution is . 1 ) ( 0 x x t e dt e x F Memory less property of the exponential distribution: The types of applications of the exponential distribution in reliability, and component or machine life time problems is influenced by the memory less or lack of memory property of the exponential distribution.

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