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MTH 265 - EXAM II REVIEW Friday, February 11 For this exam you may bring ONE 8.5 inch by 11 inch page of notes (both sides). You may put any formula on your note page, calculator command, and reminders. You may NOT put examples with solutions or any of our homework problems with solutions on your page. Inappropriate items on your formula sheet will be removed before you take your exam. 3.3 Continuous Random Variables Be able to find the cumulative distribution function for a continuous random variable. See Example 3.22. Be able to find the probability of event occurring for a continuous random variable. See Example 3.21. Be able to find the mean, variance, and standard deviation for a continuous random variable from a given experiment. See Example 3.24. Given a cumulative probability distribution function, be able to use it to find the median or any percentile. Typical and reasonable test questions would be like problems 1, 7, and 11. 3.4 Functions of Random Variables Be able to determine the mean, variance, and standard deviation of a constant multiple of a random variable (given the mean and standard deviation of the random variable). See example 3.25. Be able to determine the mean, variance, and standard deviation of a linear combination of a set of independent random variables (given the mean and standard deviation of the random variables). See example 3.27. Be able to determine the mean, variance, and standard deviation of the mean of a random sample from a population with known mean and standard deviation. See example 3.28. Typical and reasonable test questions would be like problems 1, 3, 5, and 9. 4.1 The Binomial Distribution Know what requirements must be fulfilled for a random variable to be Binomial. See p. 120. Be able to find the distribution for a Binomial random variable (recognize it is Binomial and find the mean, variance, and standard deviation). See example 4.1. Be able to calculate probabilities for a Bernoulli random variable. See example 4.2 through 4.5. Typical and reasonable test questions would be like problems 1, 3, 5, 7, and 9. 4.2 The Poisson Distribution Be able to find recognize when you have a Poisson random variable. Clue into the wording that there is a number of occurrences of an event in a specified unit (volume, time, area…). Know how to find the mean, variance, and standard deviation for a Poisson random variable. Be able to determine the parameter, , needed to specify the Poisson probability distribution. Be able to calculate probabilities for a Poisson random variable using the probability distribution or using your poissonpdf or poissoncdf features of your calculator. See examples 4.6, 4.7, 4.9, 4.10, and 4.11. In some circumstances we can use the Poisson distribution to approximate the Binomial. If I want you to demonstrate the ability to do this on the test, I will specifically tell you to use the Poisson to approximate the Binomial. See page 127 for an example of this approximation. Typical and reasonable test questions would be like problems 1, 2, 4, 5, 7 (Poisson approximation to the Binomial). 4.3 The Normal Distribution Know the proportion of any normal population that is within one, two, or three standard deviation of the mean. See page 134. Be able to calculate the area under the normal curve (probabilities) for given values of the standard normal random variable Z. See example 4.15, 4.16, and 4.17. Be able to calculate percentiles for a standard normal random variable. See example 4.18. Be able to convert any normal random variable to a standard normal random variable by using the z-score. Remember the z-score tells us how many standard deviations above or below the mean the value is. See example 4.12 and 4.13. Be able to calculate probabilities for any normal random variable given its mean and standard deviation. Be able to calculate percentiles for any normal random variable. Typical and reasonable test questions would be like problems1, 3, 5, 7, and 9. 4.5 The Exponential Distribution Know when a random variable can be categorized as an Exponential Random Variable. Know how to use the cumulative Exponential distribution to calculate probabilities. See example 4.25. Know the mean, variance, and standard deviation of an Exponential Random Variable. Pay attention to the wording difference when you are given λ versus when you are given the mean of an Exponential process. Carefully read Example 4.26 where the “mean rate of 15 particles per minute” was given and that is λ. λ is defined as a mean number of occurrences per unit. If you are given the mean time until the next occurrence or the mean lifetime, then that is the mean of the random variable and NOT λ. See Example 4.27. Don’t forget that the Exponential distribution has no memory, so asking to find the probability a component lasts two years will give the same probability as asking to find the probability that a component lasts two more years after it already lasted three years. Typical and reasonable test questions would be like problems 1, 2, 3, 4, 5, and 6. 4.8 The Central Limit Theorem Be able to use the Central Limit Theorem to determine the mean, variance, and standard deviation for the sample mean of a simple random sample and be able to calculate probabilities using them. See example 4.31. Be able to use the normal approximation to the Binomial and use the continuity correction. See Example 4.32 and 4.33. Typical and reasonable test questions would be like problems 1, 3, 5, and 7. 5.1 Point Estimation Know the vocabulary: point estimate, statistic, parameter, and bias. 5.2 Large-Sample Confidence Intervals for a Population Mean Be able to find confidence intervals for the mean of a population using a sample from a population with known mean and standard deviation. See examples 5.3, 5.2, 5.3, and 5.9. Know how to interpret a confidence interval. Read page182 and 183! See Example 5.7 and 5.8. Be able to find the level of a given confidence interval for the mean of a population. See example 5.6. Be able to find the size of the sample needed to obtain a given level of confidence for the mean of a population. See example 5.9. Know what the margin of error is and how to interpret it. Typical and reasonable test questions would be like problems 3, 5, 7, and 9. 5.3 Confidence Intervals for Proportions Be able to find two-sided confidence intervals for the proportion of successes of a population. Use the formula that appears on page 190…NOT the one in the box on page 193. See example 5.11. Be able to find the size of the sample needed to obtain a given level of confidence for the proportion of successes of a population. See examples 5.12 and 5.13. Typical and reasonable test questions would be like problems 1, 2, 3, and 13.