elemprob-fall2010-page10

elemprob-fall2010-page10 - The rst basic principle is to...

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We used the word “distribution” several times. This is used to refer to the probability that a random variable has values lying in a given set. A fuller definition will be given later, but for now we can give a partial definition. A random variable that takes only countably many values is called a discrete random variable. The number that appears when rolling a die or the flip when a coin first shows heads are examples of discrete random variables. If we spin a spinner and the value of the random variable is the angle the spinner makes with the x axis, this is not a discrete random variable, since it can take any value between 0 and 2 π . If X is a discrete random variable, we can look at the function f ( x ) = P ( X = x ). The function f maps the reals into [0 , 1], and is 0 except for countably many values. f is sometimes called the probabiloty mass function for X or the density function of X . The distribution of X is the collection of numbers P ( X A ), where A runs over the collection of subsets of the reals. 4 Permutations and combinations
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Unformatted text preview: The rst basic principle is to multiply. Suppose we have 4 shirts of 4 dierent colors and 3 pants of dierent colors. How many possibilities are there? For each shirt there are 3 possibilities, so altogether there are 4 3 = 12 possibilities. An example, How many license plates of 3 letters followed by 3 numbers are possible? Answer: (26) 3 (10) 3 , because there are 26 possibilities for the rst place, 26 for the second, 26 for the third, 10 for the fourth, 10 for the fth, and 10 for the sixth. We multiply. How many ways can one arrange a,b,c ? One can have abc, acb, bac, bca, cab, cba. There are 3 possibilities for the rst position. Once we have chosen the rst position, there are 2 possibilities for the second position, and once we have chosen the rst two possibilities, there is only 1 choice left for the third. So there are 3 2 1 = 3! arrangements. In general, if there are n letters, there are n ! possibilities. 10...
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