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Mathematics
Set Theory, Logic, Probability, Statistics
Solving 2 Problems with Exponential RVs: CLT & Probability
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[QUOTE="StoneTemplePython, post: 6777023, member: 613025"] let's tackle part 1 first the fact that you didn't use a poisson distribution here is a red flag you need to think about. You should also explicitly write out the random variable and what is stands for... jumping straight into calculations can lead to problems like this. As is you seem to think the problem involves adding 480 iid exponential random variables. I don't see how this can possibly be the case -- we are interested in 500 students not 48. Note: even if your setup was correct, there's a linearity problem -- the combined random variable would in fact have variance of 480 but would have standard deviation of $\sqrt{480}$ -- variance adds in the case of iid random variables but standard deviation does not-- you can't interchange sums of positive numbers and square roots due to negative convexity. so writing this out, what you actually want to know is whether $S_{500} \leq 480$ this is standard partial sum notation you may have seen in calculus e.g. $s_3 = x_1 + x_2 + x_3$, except we are summing random variables, $S_{500} = X_1 + X_2 + ... + X_{500}$ where each $X_j$ represents a student, having iid arrival / service time that is an exponential random variable with parameter $\lambda = 1$ so what is $Pr\big(S_{500} \leq 480\big) $ well this is asking whether the first 500 arrivals occur at some time less than or equal to 480 minutes, which is equivalent to asking whether the number arrivals at time 480 are at least 500 in a poisson process. i.e. whether $Pr\big(N(t)) \geq 500\big)$ with $t = 480$ and $N(t)$ is the 'counting' random variable which counts the number of iid exponentially distributed arrivals in $(0, t]$ and from your text you should know $Pr\big(N(t) = k )$ is precisely given by a poisson distribution. From here I'd suggest calculating the answer exactly in say excel, and then doing the normal approximation and comparing answers. = = = = = note: you didn't give any background on the course or book you are using so I had to guess on what you know. It is possible that some of what I said won't make sense. My first suggestion of completely writing out the problem and what are the random variables you're interested, and why stands in any case. Second, there are many good texts on poisson processes. These days I'd probably recommend the freely available book by Blitzstein and Hwang which has a nice chapter on Poisson processes that you may want to go through. See here: [url]https://projects.iq.harvard.edu/stat110/home[/url] [/QUOTE]
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Solving 2 Problems with Exponential RVs: CLT & Probability
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