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Precalculus Mathematics Homework Help
Statistics uniform distribution problem
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[QUOTE="Ray Vickson, post: 5093778, member: 330118"] You don't need to type much out, and you do not need to present the diagram. You just have a random variable of the form ##X = 4 \cos(\Theta)##, where ##\Theta## is uniformly distributed on ##(0, \pi/2)##. You want to compute ##EX## and ##P(X \leq 3)##. In your simple approach to ##EX## you have made the fundamental error that beginners often make: for a NONLINEAR function ##h(\theta)## we have ##E h(\Theta) \neq h(E\Theta)##, usually. Your error was to assume that was a true equation, but most often it is not. In this particular case, it is definitely not true. Think about a simple case of a discrete random variable ##Y##, taking values ##y_i## with probabilities ##p_i##; for example, ##P(Y=y_1) = P(Y = y_2) = 1/2##. For a function f(y) we have ##Ef(Y) = \sum p_i f(y_i)##, but ##f(EY) = f(\sum p_i y_i)##. Generally, you will not have ##\sum p_i f(y_i) = f(\sum p_i y_i)## unless ##f## is a linear function of the form ##f(y) = ay + b##. As you said, ##P(X \leq 3) = P(\cos(\Theta) \leq 3/4)##. Now look at the graph of ##y = \cos(\theta)## on the interval ##0 \leq \theta \leq \pi/2##. What does the region ##y \leq 0.75## look like on the ##\theta##-axis? You might wonder: why did I write ##\Theta## sometimes and ##\theta## at other times? It was not a typo: I was respecting the mostly-accepted standards of probability writing, whereby a random variable is typically denoted by an upper-case letter and its possible values by the corresponding lower case letter. So, the random variable ##\Theta## takes values ##\theta## that lie in the interval ##(0,\pi/2)##. Your book apparently did not use that convention, at least in this case. [/QUOTE]
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Precalculus Mathematics Homework Help
Statistics uniform distribution problem
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