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ESM3A HW8 - Jacobs University Bremen School of Engineering...

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Jacobs University Bremen School of Engineering and Science Peter Oswald Fall Term 2010 120201 ESM3A — Problem Set 8 Issued: 4.11.2010 Due: Wednesday 10.11.2010 (in class) This is about continuous random variables and their properties. Compare e.g. Lectures 10 and 11 in the posted script. 8.1. Which of the following are distribution functions? For those that are, give the correspond- ing density function. a) F ( x ) = ( e - 1 /x if x > 0, 0 otherwise, b) F ( x ) = e x e x + e - x , x IR, c) F ( x ) = e - x 2 + e x e x + e - x , x IR. 8.2. The median med( X ) of a continuous random variable X is the real number t that satisfies P ( X t ) = 1 / 2 (it can happen that there is a whole interval of t , in which case the median is usually defined as the midpoint of this interval). In contrast to expectations, the median always exists. Find the median of a uniform distribution on [ a, b ], of a normally distributed random variable N ( μ, σ 2 ), of an exponentially distributed random variable (parameter λ ), of a Cauchy random variable, and of the Pareto random variable (see next problem).
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