TOPIC 4 Continuous Probability Distributions

Start Thinking As a web designer you face a task, one that involves a continuous measurement of downloading time which could be any value and not just a whole number. How can you answer the following questions:As a web designer you face a task, one that involves a continuous measurement of downloading time which could be any value and not just a whole number. How can you answer the following questions:  What proportion of the homepage downloads take more than 10 seconds?  How many seconds elapse before 10% of the downloads are complete? etc.

Continuous Probability Distributions Uniform Exponential Normal Gamma Weibull Beta

Uniform Distributions Equally likely outcomesEqually likely outcomes Flat probability density functionFlat probability density function Uniform distribution probabilityUniform distribution probability Expectation and VarianceExpectation and Variance x f(x)f(x) d c a b Cumulative distribution functionCumulative distribution function

ExampleExample You’re production manager of a soft drink bottling company. You believe that when a machine is set to dispense 12 oz., it really dispenses 11.5 to 12.5 oz. inclusive. Suppose the amount dispensed has a uniform distribution. What is the probability that less than 11.8 oz. is dispensed? What are the expectation and variance of the volume dispensed?

Example Solution This is actually: Area = Height × Base f(x)f(x) x

ExerciseExercise A computer random number generator produces numbers that have a uniform distribution between 0 and 1. a)If 20 random numbers are generated, what are the expectation and variance of the number of them that lie in each of the four intervals [0.00, 0.30], [0.30, 0.50], [0.50, 0.75], [0.75, 1.00]? b)What is the probability that exactly five numbers lie in each of the four intervals? [0.00, 0.30] [0.30, 0.50] [0.50, 0.75] [0.75, 1.00]

Exercise Solution a) b)

Exponential Distributions Often used to model failure or waiting times and inter-arrival timesOften used to model failure or waiting times and inter-arrival times Probability density functionProbability density function Exponential Distribution ProbabilityExponential Distribution Probability Expectation and VarianceExpectation and Variance x Cumulative Distribution FunctionCumulative Distribution Function

ExampleExample A team of underwater salvage experts sets sail to search the ocean floor for the wreckage of a ship that is thought to be sunk within a certain area. Their boat is equipped with underwater sonar. The captain’s experience is that in similar situation it has taken an average of 20 days to locate a wreck. The time in days taken to locate the wreck can be modeled by an exponential distribution. a)The captain’s contractor has offered a bonus if it is possible to reduce searching cost by locating the wreck within the first week. What is the probability of this to be? b)The captain is only authorized to search for at most 4 weeks before calling off the search. What is the probability that the captain has to call off the search without success?

Example Solution 7 28 x Probability of bonus = 0.30 Probability search call off = 0.25

Poisson Process A Poisson process (with parameter λ) is one such process where these time intervals are independent random variables having exponential distributions with parameter λ Events Time X1X1 X2X2 X3X3 X4X4 X5X5 Independent exponential random variable

ExampleExample Customers arrive at a service window according to a Poisson process with parameter λ = 0.2 per minute. a)What is the probability that the time between two successive arrivals is less than 6 minutes? b)What is the probability that there will be exactly three arrivals during a given 10 minute period? Answer: a) a) b)λ = 0.2 per minute = 0.2 × 10 = 2.0 per 10 minutes = 0.2 × 10 = 2.0 per 10 minutes

Gamma Distributions Has applications in areas such as reliability theory, Poisson processHas applications in areas such as reliability theory, Poisson process Probability density functionProbability density function is a gamma function is a gamma function Expectation and VarianceExpectation and Variance x f(x)f(x)

ExampleExample Suppose that the engineer in charge of the car panel manufacturing process is interested in how long it will take for 20 metal sheets to be delivered to the panel construction lines. Under the Poisson process model, this time X has a gamma distribution with parameter k = 20 and λ = 1.6. What are the expectation and standard deviation of the waiting time? Answer: a) a) b) b)

Weibull Distributions Used to model failure and waiting timesUsed to model failure and waiting times Probability density functionProbability density function The cumulative distribution functionThe cumulative distribution function Expectation and VarianceExpectation and Variance x f(x)f(x)

ExampleExample A brake pad made from a new compound is tested in cars that are driven in city traffic. The random variable X, which measures the mileage in 1000-mile units that the cars can be driven before the brake pads wear out, has a Weibull distribution with parameters a = 2 and λ = a)What is the median mileage? And what does it means? b)What is the probability that a set of brake pads last longer than 10,000 miles c)If assumed that a = 2 What is the expected mileage ? Answer: a) a) About half of the brake pads will last longer than 6938 miles

Example Solution b)c)

Beta Distributions Often Used to model proportionsOften Used to model proportions Probability density functionProbability density function for 0 ≤ x ≤ 1 for 0 ≤ x ≤ 1 Expected proportion and Variance in the proportionExpected proportion and Variance in the proportion x f(x)f(x)