Continuous Distributions

1. Section 07 Continuous Distributions

2. Uniform • x has equal probability over entire interval • Parameters: • a – beginning of interval • b – end of interval

3. Normal • Parameters: • μ – mean • σ2 – variance • n – sample size • Standard normal distribution is

4. Fun things to do with the Normal Distribution • Standardizing • Given a normal random variable , find • Define • Then

5. Fun things to do with the Normal Distribution • Approximating another distribution • A random variable X with mean μ and variance σ2 is sometimes approximated by assuming the distribution of X is approximately • Often the question will ask for an approximate probability of some interval – assume normal distribution if not specifically mentioned • This is also often used for sums of random variables

6. Fun things to do with the Normal Distribution • Integer correction for the normal approximation • When estimating a discrete distribution using the normal distribution • If and are integers, to find discrete probability • We assign normal r. v. the same mean and variance as X and find • We do this because endpoints matter in discrete distributions, but not continuous

7. Exponential • x is often time between events • Parameters: • λ • Can also be described with so that • Questions can be ambiguous – look for description of mean Exponential and Poisson are often used in tandem; eg, exponential represents time until an event occurs, Poisson represents number of events occurring in a period of time. This is because they both use the variable λ.

8. Fun things to do with the Exponential Distribution • The minimum of a collection of independent exponential random variables • Suppose independent r.v. each have exponential distributions with means • If • then has an exponential distribution with mean • These types of word problems are pretty common with other distributions as well

9. Fun things to do with the Exponential Distribution • The minimum of a collection of independent exponential random variables - alternative • Suppose independent r.v. each have exponential distributions with means and • Convert to Poisson variables with means • Then add the means to get the mean of • now has a Poisson distribution with mean • This is the same as an exponential distribution with mean

10. Gamma • Parameters: • α and β • for x>0 only • for positive integers n

11. Likelihood of Distributions • DEFINITELY know • Uniform • Normal • Exponential • Try to know • Gamma

12. Actex, Ex 7-6, pg 212 A student received a grade of 80 on a math final where the mean grade was 72 and the standard deviation was s. In the statistics final, he received a 90, where the mean grade was 80 and the standard deviation was 15. If the standardized scores (i.e., the scores adjusted to a mean of 0 and standard deviation of 1) were the same in each case, find s.

13. Sample Exam, #88 The waiting time for the first claim from a good driver and the waiting time for the first claim from a bad driver are independent and follow exponential distributions with means 6 years and 3 years, respectively. What is the probability that the first claim from a good driver will be filed within 3 years and the first claim from a bad driver will be filed within 2 years? Write as an expression, not a value

14. Sample Exam, #37 The lifetime of a printer costing 200 is exponentially distributed with mean 2 years. The manufacturer agrees to pay a full refund to a buyer if the printer fails during the first year following its purchase, and a one-half refund if it fails during the second year. If the manufacturer sells 100 printers, how much should it expect to pay in refunds?