Unit12 - Unit 12 Probability and Calculus II Continuous...

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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 sub-intervals 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 sub-interval 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 dont 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 persons 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 persons 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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Unit12 - Unit 12 Probability and Calculus II Continuous...

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