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LECTURE 10ReviewConditioning “slices” the joint PDFBuffon’s Needle (1)Buffon’s Needle (2)Buffon’s Needle (3)What is a derived distribution?Why do we derive distributions?How to find them: Discrete CaseHow to find them: Continuous CaseExample 1Example 2The PDF of .LECTURE 10• Readings: Section 3.6Lecture outline• More on continuous r.v.s• Derived distributionsReviewConditioning “slices” the joint PDF• Recall the stick-breaking example:• Pictorially:Buffon’s Needle (1)• Parallel lines at distanceNeedle of length (assume )•Find P(needle intersects one of the lines).• Midpoint-nearest line distance:• Needle-lines acute angle:Buffon’s Needle (2)• Model: uniform and independent.• When does the needle intersect a line?Buffon’s Needle (3)What is a derived distribution?• It is a PMF or PDF of a function of random variables with known probability law.•Example:• Let: . Note: is a r.v.• Obtaining the PDF forinvolves deriving a distribution.Why do we derive distributions?• Sometimes we don’t need to. Example:– Computing expected values.• But often they’re useful. Examples:– Maximum of several r.v.s. (delay models)– Minimum of several r.v.s (failure models).– Sum of several r.v.s. (multiple arrivals)How to find them: Discrete Case• Consider: - a single discrete r.v.:- and a function:• Obtain probability mass for eachpossible value of :How to find them: Continuous Case• Consider: - a single continuous r.v.:- and a function:• Two step procedure:1. Get CDF of :2. Differentiate to get:• Why go to the CDF?Example 1• : uniform on•Find PDF of • Solution:1. Get the CDF:2. Differentiate:Example 2• Joan is driving from Boston to New York. Her speed is uniformly distributed between 30 and 60 mph. What is the distribution of the duration of the trip?• PDF of the velocity : •Let: • Find .The PDF of .• Use this to check that if is normal,then is also


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MIT 6 041 - Lecture Notes

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