Upon completing this module, you should be able to 1. Describe and apply the basic rules of probability (i.e., the general and special rules of addition, the general and special rules of multiplication) 2. Describe the characteristics of a binomial probability distribution 3. Demonstrate the ability to calculate or determine the probability of an outcome of a binomial situation using each of the following (where appropriate): a) the binomial probability formula and b) the binomial probability table 4. Describe the characteristics and compute the probabilities using the hypergeometric probability distribution 5. Describe appropriate conditions for use of the Poisson probability distribution and compute probabilities using this distribution 6. Describe and apply the relationship between the mean and the standard deviation of a normally distributed frequency distribution 7. Describe and apply the relationship between basic probability and a normally distributed frequency distribution, using the tables of areas under the normal curve 8. Apply the normal distribution procedures in determining the probability of certain outcomes under the above circumstances 2. Tips for success You'll write an exam at the end of this module. Please review Booking Exams now. Plan ahead to meet this requirement of this course.
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Understanding Probability Rules: The basic rules of probability include the general and special rules of addition and multiplication. The general rule of addition states that the probability of the occurrence of either of two mutually exclusive events is the sum Show more…
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Find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine if the events are unusual. If convenient, use the appropriate probability table or technology to find the probabilities. The mean number of oil tankers at a port city is 10 per day. The port has facilities to handle up to 12 oil tankers in a day. Find the probability that on a given day, (a) ten oil tankers will arrive, (b) at most three oil tankers will arrive, and (c) too many oil tankers will arrive. (a) P(ten oil tankers will arrive) = (Round to four decimal places as needed.) (b) P(at most three oil tankers will arrive) = (Round to four decimal places as needed.) (c) P(too many oil tankers will arrive) = (Round to four decimal places as needed.) A. The event in part (a) is unusual. B. The event in part (b) is unusual. C. The event in part (c) is unusual. D. None of the events are unusual.
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Find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine if the events are unusual. If convenient, use the appropriate probability table or technology to find the probabilities. The mean number of oil tankers at a port city is 15 per day. The port has facilities to handle up to 18 oil tankers in a day. Find the probability that on a given day: (a) fifteen oil tankers will arrive, (b) at most three oil tankers will arrive, and (c) too many oil tankers will arrive. (a) P(fifteen oil tankers will arrive) = (b) P(at most three oil tankers will arrive) = (c) P(too many oil tankers will arrive) = A. The event in part (a) is unusual. B. The event in part (b) is unusual. C. The event in part (c) is unusual. D. None of the events are unusual.
Let $X$ be a discrete rv with possible values $0,1,2, \ldots$ or some subset of these. The function $\Psi(s)=E\left(s^{X}\right)=\sum_{x=0}^{\infty} s^{x} \cdot p(x)$ is called the probability generating function $(\mathbf{p g f})$ of $X$ (a) Suppose $X$ is the number of children born to a family, and $p(0)=.2, p(1)=.5,$ and $p(2)=$ . $3 .$ Determine the pgf of $X .$ (b) Determine the pgf when $X$ has a Poisson distribution with parameter $\mu .$ (c) Show that $\psi(1)=1$ (d) Show that $\psi^{\prime}(0)=p(1) .$ (You'll need to assume that the derivative can be brought inside) the summation, which is justified.) What results from taking the second derivative with respect to $s$ and evaluating at $s=0 ?$ The third derivative? Explain how successive differentiation of $\psi(s)$ and evaluation at $s=0$ "generates the probabilities in the distribution." Use this to recapture the probabilities of (a) from the pgf. [Note: This shows that the pgf contains all the information about the distribution-knowing $\psi(s)$ is equivalent to knowing $p(x) . ]$
Discrete Random Variables and Probability Distributions
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