4-Chapter 7&8 - 9/26/2009 Chapter 7 Random Random...

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9/26/2009 1 Random Variables and Discrete probability Chapter 7 Distributions 7.2 Random Variables and Probability Distributions ± A random variable is a function or rule that assigns a numerical value to each simple event in a sample space. ± A random variable reflects the aspect of a random experiment that is of interest for us. ± There are two types of random variables: ± Discrete random variable ± Continuous random variable. ± A random variable is discrete if it can assume a countable number of values. ± A random variable is continuous if it can assume an uncountable number of values. Continuous random variable Discrete random variable Discrete and Continuous Random Variables 0 1 1/2 1/4 1/16 After the first value is defined the second value, and any value thereafter are known. Therefore, the number of values is countable After the first value is defined, any number can be the next one Therefore, the number of values is uncountable 0 1 2 3 . ..
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9/26/2009 2 Discrete Distributions ± In this chapter we will be dealing with discrete, in the next chapter (8) we will discuss continuous ± A table, formula, or graph that lists all possible values a discrete random variable can assume, together with associated probabilities, is called a discrete probability distribution . ± To calculate the probability that the random variable Discrete Probability Distribution To calculate the probability that the random variable X assumes the value x, P( X = x), ± add the probabilities of all the simple events for which X is equal to x, or ± Use probability calculation tools (tree diagram), ± Apply probability definitions ± If a random variable can assume values x i , then the following must be true: x a fo 1 p(x 1 Requirements for a Discrete Distribution 1 ) x ( p . 2 all for ) 0 . i x al i i i = These are the rules of probability
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9/26/2009 3 Distribution and Relative Frequencies ± In practice, often probability distributions are estimated from relative frequencies. ± Example 7.1 on pg. 212 ± A survey reveals the following frequencies (1,000s) for the number of color TVs per household. Number of TVs Number of Household x p(x Number of Households p(x) 0 1,218 0 1218/Total = .012 1 32,379 1 .319 2 37,961 2 .374 3 19,387 3 .191 4 7,714 4 .076 5 2,842 5 .028 Total 101,501 1.000 Determining Probability of Events ± The probability distribution can be used to calculate the probability of different events ± Example 7.1 – continued Calculate the probability of the following events: ± P(The number of color TVs is 3) = P(X=3) =.191 ± P(The number of color TVs is two or more) = P(X 2)=P(X=2)+P(X=3)+P(X=4)+P(X=5)= .374 +.191 +.076 +.028 = .669 ± P(The number of color TVs is two or less)= P(X< 2)=P(X=2)+P(X=1)+P(X=0)= .012+.319+ .374=.705 ± Probability calculation techniques can be used to develop probability distributions ± Example 7.2 on pg. 213 ± A mutual fund sales person knows that there is 20% chance of closing a sale on each call she makes.
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This note was uploaded on 10/14/2010 for the course ADMS adms 3333 taught by Professor Adms during the Spring '10 term at York University.

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4-Chapter 7&amp;8 - 9/26/2009 Chapter 7 Random Random...

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