All probability distributions whose variable extent depends on the parameters do not belong to the exponential family
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![All probability distributions whose variable
extent depends on the parameters do not
belong to the exponential family
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or
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- A toy factory produces battery operated toy robots for toddlers. Previous research shows that the lifetimes of these robots are Exponentially distributed with mean 6 months on regular use. A retailer buys 15 of these robots to sell. What is the probability that exactly 5 of these robots will last at least 8 months?On average, TV crash after 17,000 hours of continuous use. Although the time for the TV to crash is random, researchers have found that the duration until the computer crashes follows an exponential probability distribution. a. What is the probability that the TV will fail with a usage time of less than 17,000 hours?b. What is the probability that the TV will crash with a usage time of more than 17,000 hours?c. What is the probability that no TV will fail after more than 17,000 hours of use?d. What do you think about the results of points b and c above! What is the difference?The manager of a gas station has observed that the times required by drivers to fill their car's tank and pay are quite variable. In fact, the times are exponentially distributed with a mean of 6.5 minutes. What is the probability that a car can complete the transaction in less than 6 minutes?
- The following graph plots the fitted probabilities of default (y axis) depending on the credit balance (x axis) with the linear probability model (left) and the logistic model (right). Which alternative is NOT correct 500 1000 1500 2000 2500 500 1000 1500 2000 2500 Balance Balance O a. There will always be a balance level such that the probability of default according to the linear probability model is above 100% O b. The logit model supposes that the error term of the latent variable follows a logistic distribution O C. The linear probability model estimates probabilities of default below 0% Od. The probability of default lower than 50% is a frequent feature of linear probability models e. The logistic model estimates probabilities between 0% and 100% 00 80 90 YO 0 Probability of Default R'O 90 o zo o'o Probability of DefaultAs a data analyst, I received requests at a rate of 1 every 25 minutes, exponentially distributed. Determine the probability that the interval between succeedingmessages is more than 1 hourPlease use excel formulas when possible: You have a device that uses a single battery, and you operate this device continuously, never turning it off. Whenever a battery fails, you replace it with a brand new one immediately. Suppose the lifetime of a typical batteryhas an exponential distribution with mean 205 minutes. Suppose you operate the device continuously for three days, making battery changes when necessary. Find the probability that you will observe at least 25 failures. (Hint: The number of failures is Poisson distributed.) In the previous problem, we ran the experiment for a certain number of days and then asked about the number of failures. In this problem, we take a different point of view. Suppose you operate the device, starting with a new battery, until you have observed 25 battery failures. What is the probability that at least 15 of these 25 batteries lived at least 3.5 hours? (Hint: Each lifetime is exponentially distributed
- A microprocessor that controls the tuner in color TVs fails completely at random (that is, according to the exponential distribution). Suppose that the likelihood that a microprocessor that has survived for k years fails in year k + 1 is .0036. What is the cumulative distribution function of the time until failure of the microprocessor?The useful life of Johnson rods for use in a particular vehicle follows an exponential distribution with an average useful life of 5.2 years.You have a three-year warranty on your vehicle’s Johnson rod. What is the probability that the Johnson rod doesn’t fail before then? That is, what is the probability that its useful life doesn’t end before three years?b.If the vehicle manufacturer wants to limit the number of claims on the three-year warranty to 20%, what should the average useful life of the Johnson rod be?Discuss the connection between the Poisson and exponential distributions.
- The time between failures of our video streaming service follows an exponential distribution with a mean of 40 days. Our servers have been running for 17 days, What is the probability that they will run for at least 97 days? (clarification: run for at least another 80 days given that they have been running 17 days). Report your answer to 3 decimal places.Suppose 5 students are going to take a test independently from each other and that the number of minutes that any student needs to finish the exam has an exponential distribution with mean 80.If the test starts at 9 a.m., determine the probability that at At least one of the students finishes the exam before 9:40 am.The life X (in hours) of a battery in constant use is a random variable with exponential density. What is the probability that the battery will last more than 12 hours (h) if the average life is 8 h?
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