4.8: Chapter Key Items
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| Key Terms | Definition |
|---|---|
| Bernoulli Trials |
an experiment with the following characteristics:
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| Binomial Experiment |
a statistical experiment that satisfies the following three conditions:
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| Binomial Probability Distribution |
a discrete random variable (RV) that arises from Bernoulli trials; there are a fixed number, \(n\), of independent trials. “Independent” means that the result of any trial (for example, trial one) does not affect the results of the following trials, and all trials are conducted under the same conditions. Under these circumstances the binomial RV \(X\) is defined as the number of successes in n trials. The mean is \(\mu=n p\) and the standard deviation is \(\sigma=\sqrt{n p q}\). The probability of exactly x successes in \(n\) trials is \(P(X=x)=\left(\begin{array}{l}{n} \\ {x}\end{array}\right) p^{x} q^{n-x}\). |
| Geometric Distribution |
a discrete random variable (RV) that arises from the Bernoulli trials; the trials are repeated until the first success. The geometric variable X is defined as the number of trials until the first success. The mean is \(\mu=\frac{1}{p}\) and the standard deviation is \(\sigma = \sqrt{\frac{1}{p}\left(\frac{1}{p}-1\right)}\). The probability of exactly x failures before the first success is given by the formula: \(P(X=x)=p(1-p)^{x-1}\) where one wants to know probability for the number of trials until the first success: the \(x\)th trail is the first success. An alternative formulation of the geometric distribution asks the question: what is the probability of \(x\) failures until the first success? In this formulation the trial that resulted in the first success is not counted. The formula for this presentation of the geometric is: \(P(X=x)=p(1-p)^{x}\) The expected value in this form of the geometric distribution is \(\mu=\frac{1-p}{p}\) The easiest way to keep these two forms of the geometric distribution straight is to remember that p is the probability of success and \((1−p)\) is the probability of failure. In the formula the exponents simply count the number of successes and number of failures of the desired outcome of the experiment. Of course the sum of these two numbers must add to the number of trials in the experiment. |
| Geometric Experiment |
a statistical experiment with the following properties:
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| Hypergeometric Experiment |
a statistical experiment with the following properties:
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| Hypergeometric Probability |
a discrete random variable (RV) that is characterized by:
1. A fixed number of trials.
We sample from two groups of items when we are interested in only one group. \(X\) is defined as the number of successes out of the total number of items chosen. |
| Poisson Probability Distribution |
a discrete random variable (RV) that counts the number of times a certain event will occur in a specific interval; characteristics of the variable: The probability that the event occurs in a given interval is the same for all intervals. The events occur with a known mean and independently of the time since the last event. The distribution is defined by the mean \(\mu\) of the event in the interval. The mean is \(\mu = np\). The standard deviation is \(\sigma=\sqrt{\mu}\). The probability of having exactly \(x\) successes in \(r\) trials is \(P(x)=\frac{\mu^{x} e^{-\mu}}{x !}\). The Poisson distribution is often used to approximate the binomial distribution, when \(n\) is “large” and \(p\) is “small” (a general rule is that \(np\) should be greater than or equal to 25 and \(p\) should be less than or equal to 0.01). |
| Probability Distribution Function (PDF) | a mathematical description of a discrete random variable (RV), given either in the form of an equation (formula) or in the form of a table listing all the possible outcomes of an experiment and the probability associated with each outcome. |
| Random Variable (RV) |
a characteristic of interest in a population being studied; common notation for variables are upper case Latin letters \(X, Y, Z\),...; common notation for a specific value from the domain (set of all possible values of a variable) are lower case Latin letters \(x, y\), and \(z\). For example, if \(X\) is the number of children in a family, then \(x\) represents a specific integer 0, 1, 2, 3,.... Variables in statistics differ from variables in intermediate algebra in the two following ways. The domain of the random variable (RV) is not necessarily a numerical set; the domain may be expressed in words; for example, if \(X =\) hair color then the domain is {black, blond, gray, green, orange}. We can tell what specific value x the random variable \(X\) takes only after performing the experiment. |