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Unformatted text preview: Unit 12 Probability and Calculus II Continuous Distributions So far, we have only been concerned with discrete random variables. Re call that a discrete random variable is one that can take any one of (only) a finite number of values. In this section, we study another kind of random variable, called a continuous random variable. Definition 12.1. A Continuous Random Variable is one which may assume any of an infinite number of values on some interval of the real number line. Because a continuous R.V. can have any of an infinite number of values, events are defined as subintervals in which the value of the R.V. could fall (i.e. we never define events as specific values of the R.V.). Example 1 . Consider the experiment: randomly pick a number between 0 and 1 inclusive. Let X be the number picked, so that X is a continuous random variable. Identify the sample space, S , for this experiment and some events on that sample space. Solution: Here, X can take on any of the infinitely many numbers which lie in the interval [0 , 1]. So it is this interval that is the sample space, i.e., S = { ≤ X ≤ 1 } or S = { x  ≤ x ≤ 1 } . Notice: The notation { x  “some condition” } says “the set of all x ’s such that the specified condition is satisfied”. That is, the  means “such that”. Any subinterval of the sample space is a possible event. For instance, we might be interested in the event A , that the number chosen is bigger than .5, or in event B , that a number between 1 4 and 1 3 is chosen. We have A = { . 5 < X ≤ 1 } and B = braceleftbig 1 4 ≤ X ≤ 1 3 bracerightbig . 1 For a continuous random variable, because there are so many possible values, the probability of any exact particular value occurring is effectively 0. That is: Theorem: For a continuous R.V., X , prob { X = x } = 0 for all values x . (This is why we don’t bother with events which correspond to specific values of X .) Example 2 . Identify the following random variables as either Discrete or Con tinuous. (a) X is the sum of 2 dice when rolled ; (b) Y is the height of a randomly selected calculus student. Solution: (a) This is a discrete random variable because there are only 11 possible values (the integers 2 through 12). (b) A person’s height, when measured precisely, can have any real value within some range. That is, we could identify limits on the possible height values which might be observed  an interval outside of which a person’s height could not possibly be. For instance, it would appear reasonable to assume that no calculus student could ever be less that 1 foot tall, or more than 10 feet tall. However, within a certain range, any value could be ob served. So in this case, Y is a continuous random variable....
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 Fall '09
 OLDS,VICKY
 Calculus, Normal Distribution, Probability, Probability distribution, Probability theory, kxdx

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