ch3 - Stat 0302B Business Statistics Spring 2010-2011...

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Stat 0302B Business Statistics Spring 2010-2011 Chapter III Random Variable, Binomial and Normal Distributions § 3.1 Random Variables A variable is a random variable if the value that it assumes, corresponding to the outcome of an experiment, is a chance or random event. It is often denoted by a capital letter, e.g. X . Quantitative random variables are classified as either discrete or continuous , according to the values that X may assume. Discrete random variables generally involve “counting”, while continuous random variables generally involve “measuring”. Example 3.1 Discrete random variables: = X number of 6’s when a die is rolled 4 times = X number of telephone calls an operator will receive in the next hour = X number of accidents at a particular black traffic spot = X number of defects on a randomly selected piece of furniture = X daily production in a manufacturing plant Continuous random variables: = X height of a randomly selected college student = X length of life of a particular electronic device = X fuel-efficient (miles per gallon) of a vehicle = X total annual income of a household selected at random P.49
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Stat 0302B Business Statistics Spring 2010-2011 Note that we usually let our random variable be X or Y because of our mathematical backgrounds but any letter of the alphabet, like W or U or V or A or B , can be used as well. The probability distribution for a discrete random variable X is a formula, table, or graph that provides , the probability associated with each of the values of X . () x p Example 3.2 Consider an experiment that consists of tossing three coins, and let X equal the number of heads observed. The following table lists all the possible outcomes and the corresponding values of X : Event Coin 1 Coin 2 Coin 3 Probability X 1 E H H H 8 1 3 2 E H H T 8 1 2 3 E H T H 8 1 2 4 E H T T 8 1 1 5 E T H H 8 2 6 E T H T 8 1 1 7 E T T H 8 1 1 8 E T T T 8 0 The event 1 is associated with the simple event “observe heads on all three coins”, we assign it the value . Similarly, we assign E 3 = X 2 = X to event 2 , which is associated with the simple event “observe heads on coin 1 and coin 2, and a tail on coin 3”, and so on. The probability of each value of X can be calculated by adding the probabilities of the simple events in that numerical event, as shown in the following table. E X Events ( ) x p 0 8 E 8 1 7 6 4 , , E E E 8 3 2 5 3 2 , , E E E 8 3 3 1 E 8 Observe that . 1 = x p P.50
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Stat 0302B Business Statistics Spring 2010-2011 § 3.2 Expected Value, Mean and Variance The probability distribution in Example 3.2 can be presented graphically of the following relative frequency histogram. Just as we calculated the mean and standard deviation for a dataset of n measurements to measure the location and the variability of the data, we can also calculate a mean and standard deviation to describe the probability distribution for a random variable X . the population mean, which measures the average value of X in population, is also called the expected value of the random variable X .
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ch3 - Stat 0302B Business Statistics Spring 2010-2011...

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