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Discrete and Continuous Probability Chapters 3 and 4 I. Introduction to Probability Distributions and Random Variables (3.1). A. What is a random variable? B. What are the basic requirements of a discrete probability distribution? 1. 0P ≤ P (x) ≤ 1.0 P (x) = 1.0 2. all x C. What is a cumulative distribution? D. What is the difference between a discrete and a continuous random variable? E. What are the basic requirements of a continuous probability density? 1. height ≥ 0 2. Area under density = 1.0 II. Expected Values (3.2) A. What is meant by a expected value and how is it calculated? B. What is meant by a expected value of a function (of a random variable) and how is it calculated? 1. How is the population mean related to expected value? 2. How is the population variance related to expected value? 3. If y = a + bx, where a and b are constants, what is the formula for the: a. mean of y b. variance of y. C. When would you use Chebychev’s Theorem to compute probabilities and why? III. Discrete probability Distributions A. The Bernoulli Distribution. 1. What is the Bernoulli distribution? 2. What conditions/situations does it describe? 3. What is the mean of a Bernoulli random variable? 4. What is the variance of a Bernoulli random variable? B. The Binomial Distribution 1. What is the Binomial distribution? 2. What conditions/situations does it describe? 3. What is the mean of a binomial random variable? 4. What is the variance of a binomial random variable? C. The Poisson Distribution 1. What is the Poisson distribution? 2. What conditions/situations does it describe? 3. What is the mean of a poisson random variable? 4. What is the variance of a poisson random variable? D. The Geometric Distribution 1. What is the geometric distribution? 2. What conditions/situations does it describe? 3. What is the mean of a geometric random variable? 4. What is the variance of a geometric random variable? IV. Continuous Distributions A. What is the primary difference between a continuous and a discrete distribution? B. What is the probability that a continuous random variable takes a particular (exact) value? C. The Uniform Distribution 1. What is the uniform distribution? 2. What conditions/situations does it describe? 3. What is the mean of a uniform random variable? 4. What is the variance of a uniform random variable? V. The standard normal distributions and its tables (4-1 to 4-3) A. The table displays areas between the mean (0) and positive values of z. Remember that the probability that a random variable is less than a particular value a is written F (a), where F (a) is the cumulative distribution. The standard normal table then displays T able Area = F (z) − .5 z > 0. B. You should be able to use the table to compute the Pr{0 < z < b} for a given b > 0. C. You should be able to use the table to compute the Pr{z > b}. D. You should be able to use the table to compute the Pr{a < z < b} . E. You should be able to use the table to compute the Pr{a > z or z > b}. F. You should also be able to Þnd a z which produces a certain probability. VI. Transforming a normal random variable into a z-score and vice versa (4-4 to 4-5). A. There are two important formula in this section: 1. The Þrst formula transforms a normal x into a z-score x − µx z= σx 2. The second formula transforms a z-score into a normally distributed x, x = µx + zσ x . B. Using these formulae, you should be able to make any probability statements from section 4-3 as they apply to x (the non-standard normal). VII. The normal distribution as an approximation to other distributions (4-6) This section illustrates a principle that will be generalized in the following chapter (the Central Limit Theorem). That principle states that many distributions can be approximated using the normal distribution if n (the sample size) is large. The Þrst such distribution we look at is the binomial. A. To approximate binomial probabilities: 1. Apply the continuity correction if 20 ≤ n ≤ 50. √ 2. Compute µx = np and σx = npq to transform the binomial variables to z-scores using (1) . (1) (2)