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Random Variable A random variable X is a rule that assigns a numerical value to each outcome in the sample space of an experiment. A discrete random variable can take on specific, isolated numerical values, like the outcome of a roll of a die, or the number of dollars in a randomly chosen bank account. A continuous random variable can take on any values within a continuum or an interval, like the temperature in Central Park, or the height of an athlete in centimeters. Discrete random variables that can take on only finitely many values (like the outcome of a roll of a die) are called finite random variables. Probability Distribution The probability P (X = x) is the probability of the event that X = x. Similarly, the probability that P (a
These probabilities may be estimated, empirical, or abstract For a finite random variable, the collection of numbers P (X = x) as x varies is called the probability distribution of X, and it is useful to graph the probability distribution as a histogram. Bernoulli Trials and the Binomial Distribution A Bernoulli trial is an experiment with two possible outcomes, called success and failure. Each outcome has a specified probability: p for success and q for failure (so that p+q = 1). If we perform a sequence of n independent Bernoulli trials, then some of them result in success and the rest of them in failure. The probability of exactly x successes in such a sequence is given by P (exactly x successes in n trials) = C (n, x) plan-x. If X is the number of successes in a sequence of n independent Bernoulli trials, with probability p for success and q for failure, then X is said to have a binomial distribution.
This distribution is given by the above formula P (X = x) = C (n, x) plan-x for x running from 0 to n. Measures of Central Tendency: Mean, Median, and Mode of a Set of Data A collection of specific values, or "scores " xn of a random variable X is called a sample. If {xn} is a sample, then the sample mean of the collection is x = x 1 + x 2 +... + xn n = xi n, where n is the sample size: the number of scores. The sample median m is the middle score (in the case of an odd-size sample), or average of the two middle scores (in the case of an even-size sample), when the scores in a sample are arranged in ascending order.
A sample mode is a score that appears most often in the collection. (There may be more than one mode in a sample. ) If the sample we are using consists of all the values of X from an entire population (for instance, the SAT of every graduating high school student who took the test), we refer to the mean, median, and mode above as the population mean, median, and mode. Ch 7 Journal Entry An experiment is an occurrence whose result, or outcome is uncertain. The set of all possible outcomes is called the sample space for the experiment. Given a sample space S, an event E is a subset of S. The outcomes in E are called the favorable outcomes. We say that E occurs in a particular experiment if the outcome of that experiment is one of the elements of E, that is, if the outcome of the experiment is favorable.
Combining Events If E and F are events In an experiment, then: E' is the event that E does not occur. E F is the event that either E occurs or F occurs (or both). E F is the event that both E and F occur. E and F are said to be disjoint or mutually exclusive if (E F) is empty. Estimated Probability If an experiment is performed N times, and the event E occurs fr (E) times, then the ratio P (E) = fr (E) N is called the relative frequency or estimated probability of E. The number fr (E) is called the frequency of E.
N, the number of times that the experiment is performed, is called the number of trials or the sample size. If E consists of a single outcome s, then we refer to P (E) as the estimated probability of the outcome s, and write P (s).
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