280wk10_x4 - Averaging and Expectation Suppose you toss a...

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Unformatted text preview: Averaging and Expectation Suppose you toss a coin thats biased towards heads (Pr(heads) = 2 / 3) twice. How many heads do you expect to get? In mathematics-speak: Whats the expected number of heads? What about if you toss the coin k times? Whats the average weight of the people in this class- room? Thats easy: add the weights and divide by the num- ber of people in the class. But what about if I tell you Im going to toss a coin to determine which person in the class Im going to choose; if it lands heads, Ill choose someone at random from the first aisle, and otherwise Ill choose someone at random from the last aisle. Whats the expected weight? Averaging makes sense if you use an equiprobable distri- bution; in general, we need to talk about expectation . 1 Random Variables To deal with expectation, we formally associate with ev- ery element of a sample space a real number. Definition: A random variable on sample space S is a function from S to the real numbers. Example: Suppose we toss a biased coin (Pr( h ) = 2 / 3) twice. The sample space is: hh - Probability 4 / 9 ht - Probability 2 / 9 th - Probability 2 / 9 tt - Probability 1 / 9 If were interested in the number of heads, we would con- sider a random variable # H that counts the number of heads in each sequence: # H ( hh ) = 2; # H ( ht ) = # H ( th ) = 1; # H ( tt ) = 0 Example: If were interested in weights of people in the class, the sample space is people in the class, and we could have a random variable that associates with each person his or her weight. 2 Probability Distributions If X is a random variable on sample space S , then the probability that X takes on the value c is Pr( X = c ) = Pr( { s S | X ( s ) = c } ) Similarly, Pr( X c ) = Pr( { s S | X ( s ) c } . This makes sense since the range of X is the real numbers. Example: In the coin example, Pr(# H = 2) = 4 / 9 and Pr(# H 1) = 5 / 9 Given a probability measure Pr on a sample space S and a random variable X , the probability distribution asso- ciated with X is f X ( x ) = Pr( X = x ). f X is a probability measure on the real numbers. The cumulative distribution associated with X is F X ( x ) = Pr( X x ). 3 An Example With Dice Suppose S is the sample space corresponding to tossing a pair of fair dice: { ( i, j ) | 1 i, j 6 } . Let X be the random variable that gives the sum: X ( i, j ) = i + j f X (2) = Pr( X = 2) = Pr( { (1 , 1) } ) = 1 / 36 f X (3) = Pr( X = 3) = Pr( { (1 , 2) , (2 , 1) } ) = 2 / 36 . . . f X (7) = Pr( X = 7) = Pr( { (1 , 6) , (2 , 5) , . . . , (6 , 1) } ) = 6 / 36 . . . f X (12) = Pr( X = 12) = Pr( { (6 , 6) } ) = 1 / 36 Can similarly compute the cumulative distribution: F X (2) = f X (2) = 1 / 36 F X (3) = f X (2) + f X (3) = 3 / 36 ....
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280wk10_x4 - Averaging and Expectation Suppose you toss a...

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