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lecture7 - Stat 302 Introduction to Probability Jiahua Chen...

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Unformatted text preview: Stat 302, Introduction to Probability Jiahua Chen January-April 2011 Jiahua Chen () Lecture 7 January-April 2011 1 / 24 Buffon’s needle Consider a random experiment in which a stick of unit length is randomly thrown onto a floor painted with parallel lines at unit distance apart. One may describe the possible outcome by two measures: (1) the distance of the center of the stick to the nearest line; (2) the angle between this stick and the nearest line (take the acute one). The sample space is hence Ω = { ( d , θ ) : ≤ d ≤ 0.5; 0 ≤ θ ≤ π / 2 } . A figure will be drawn on the blackboard. Jiahua Chen () Lecture 7 January-April 2011 2 / 24 Buffon’s needle Because Ω contains infinite number of distinct sample points, it is not possible to assign a positive probability to each of them without violating Axiom P ( Ω ) = 1. Instead, for each event A ⊂ Ω , we define P ( A ) = area of A area of Ω . This definition can be verified to satisfy three Axioms for a probability measure. Jiahua Chen () Lecture 7 January-April 2011 3 / 24 Buffon’s needle Based on this model, what is the probability that the stick crosses a line? Let A be the event that the stick crosses a line. It is seen that A = { ( d , θ ) : 0.5 sin ( θ ) > d } . The area of A is given by integraldisplay π / 2 0.5 sin ( θ ) d θ = 0.5 and the area of Ω = π / 4. Hence, P ( A ) = 2 / π . Jiahua Chen () Lecture 7 January-April 2011 4 / 24 Buffon’s needle Based on this model, what is the probability that the stick crosses a line? Let A be the event that the stick crosses a line. It is seen that A = { ( d , θ ) : 0.5 sin ( θ ) > d } . The area of A is given by integraldisplay π / 2 0.5 sin ( θ ) d θ = 0.5 and the area of Ω = π / 4. Hence, P ( A ) = 2 / π . When you are bored, throw a stick a few thousand of times to see if you can get an accurate value of π . Jiahua Chen () Lecture 7 January-April 2011 4 / 24 Learn more from this example Unlike most examples in previous lectures, this random experiment has a “continuous” sample space. We have to assign ZERO probability to subsets with zero area. For instance, if B is event that the angle between the stick and the lines is π / 6. It is seen that P ( B ) = 0. Yet B is not an empty event. In addition, the sample space is the union of events of various angles. The probability of the sample space is 1, but the event with each specific angle is 0. Jiahua Chen () Lecture 7 January-April 2011 5 / 24 Continuous random variable Take the same sample space. Let X be the angle between the stick to the lines. Apparently, X is a function defined on Ω , and its range is a continuous interval [ 0, π / 2 ] ....
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lecture7 - Stat 302 Introduction to Probability Jiahua Chen...

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