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Chapter4 - Chapter 4 lecture notes Math 431 Spring 2011...

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Chapter 4 lecture notes Math 431, Spring 2011 Instructor: David F. Anderson Chapter 4: Random Variables Section 4.1 Oftentimes, we are not interested in the specific outcome of an experiment. Instead, we are interested in a function of the outcome . Example 1. Consider rolling a fair die twice. S = { ( i, j ) : i, j ∈ { 1 , . . . , 6 }} . Suppose we are interested in computing the sum, i.e. we have placed a bet at a craps table. Let X be the sum. Then X ∈ { 2 , 3 , . . . , 12 } is random as it depends upon the outcome of the experiment. It is a random variable . We can compute probabilities associated with X . P ( X = 2) = P { (1 , 1) } = 1 / 36 P ( X = 3) = P { (1 , 2) , (2 , 1) } = 2 / 36 P ( X = 4) = P { (1 , 3) , (2 , 2) , (1 , 3) } = 3 / 36 . Can write succinctly Sum, i 2 3 4 5 6 . . . 12 P ( X = i ) 1 / 36 2 / 36 3 / 36 4 / 36 5/36 · · · 1 / 36 Example 2. Let X denote the number of successes in n independent trials if the probability of success on each is 0 < p < 1. We computed this last class. We have that for k ∈ { 0 , . . . , n } P { X = k } = n k p k (1 - p ) n - k . Definition 1. Let S be a sample space. Then a function X : S R is a random variable. 1
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Example 3. Independent trials, with success probability p , are being conducted. Let X be the number of the trial (flip) which results in the first success. What is P { X = k } , for k 1? Solution: Note that P { X = 1 } = p P { X = 2 } = (1 - p ) p . . . P { X = k } = (1 - p ) k - 1 p. This should satisfy k =1 P { X = k } = 1, right? Does it? X k =1 P { X = k } = X k =1 (1 - p ) k - 1 p = p X k =0 (1 - p ) k = p 1 1 - (1 - p ) = 1 . Later we will recognize this RV as a geometric random variable . Distribution Function. For a random variable X , we define the distribution function , or cumulative distribution function , written F X ( x ) or F ( x ), by F X ( x ) = F ( x ) = P { X x } . A few observations: 1. If a b , then { X a } ⊂ { X b } . Therefore, for a b , F ( a ) = P { X a } ≤ P { X b } = F ( b ) , and we see that the distribution function of a random variable is non-decreasing. 2. F is defined for all x R . We will see other properties of F in Section 4.10. Example 4. Consider rolling a die. Let S = { 1 , 2 , 3 , 4 , 5 , 6 } . Let X : S R be given by X ( s ) = s 2 . Then, P ( X = i ) = 1 / 6 for each i ∈ { 1 , 4 , 9 , 16 , 25 , 36 } . Note, however, that F is defined for all t R . So function is constant between numbers and F (1) = P ( X 1) = P ( X = 1) = 1 / 6 , F (4) = P ( X 4) = P ( X = 1 or X = 4) = P ( X = 1) + P ( X = 4) = 2 / 6 , etc DRAW PLOT. 2
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Section 4.2: Discrete Random Variables Consider a random variable, X : S R . Let R ( X ) be the range of X . If R ( X ) is finite or countably infinite, then X is said to be discrete. For a discrete random variable X , we define the probability mass function p ( a ) or p X ( a ) of X by p ( a ) = P { X = a } . If R ( X ) = { x 1 , x 2 , . . . } is the range of X , then p ( x i ) 0 , for i = 1 , 2 , . . . p ( x ) = 0 , for all other values of x. Because X must take one of those values we also see that X i =1 p ( x i ) = 1 . Often helpful to present probability mass functions graphically. Give some examples. Maybe I : p (0) = 1 7 , p (1) = 2 7 , p (2) = 1 7 , p (3) = 3 7 . See pages 123 and 124 for more examples (like rolling two die).
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