I would like to ask below question: i am trying to detect a kind of event by experiments, i repeat the experiment for 21 times, but no event of interest occurred. My quesiton is based on current results (21 repeat without any occurrence), what is the reasonable estimation of probability of the event? (Or, posterior desitribution of the probaility of the event?) Looking forward to your great ideas.
15147
What is the best technique to calculate a confidence interval of a binomial experiment, if your estimate is that $p=0$ (or similarly $p=1$) and sample size is relatively small, for example $n=25$?
80386
I'm not a statistician. So, please bear with my blunders if any.  Would you please explain in a simple manner how simulation is done? I know that it picks some random sample from a normal distribution and use for simulating. But, don't understand clearly.
89536
Having studied parametric Bayesian statistics during the two last years, I plan to begin to self-study non parametric Bayesian model during this summer and look for recommendations. I would like the book(s) to cover both the theoritical aspects and the practical ones (with implementation, examples of models ...) Thanks for your recommendations.
105816
If N Bayesians all estimate the probability of some event, how do you aggregate their estimates to get a final estimate? (Assume they are all equally trustworthy, or at least you have no reason to favor one over another)  For example, if one says 50%, another 80%, and a third 70%, do you take the mean, median, geometric mean?
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I did a science PhD doing applied bayesian statistics, and I liked it so much that I decided to shift to statistics.  For the last 2-3 years have been working at a firm that has more of a frequentist focus.  It seems to me that the bayesian approach (at least what I learned) has an underlying theory that was laid out and I could understand.  Now I find this - more classical - field fascinating but I am looking for some resources to get a unified view of the matter.  For example, I am learning and applying anovas, F-tests and chi-squares, thinking about degrees of freedom, etc but part of me is looking for deeper understanding that would make it more than just methods.  Does anyone have any suggestions on books or more resources that provide such a unified, deeper focus?
46686
Lets say I have three independent sources and each of them make predictions for the weather tomorrow. The first one says that the probability of rain tomorrow is 0, then the second one says that the probability is 1, and finally the last one says that the probability is 50%. I would like to know the total probability given that information.  If apply the multiplication theorem for independent events I get 0, which doesn't seem correct. Why is not possible to multiply all three if all sources are independent? Is there some Bayesian way to update the prior as I get new information?  Note: This is not homework, is something that I was thinking about.
17973
I recently came across the paper . The author raised a few common misconceptions regarding hypothesis testing and p-values, about which I have two specific questions:   The p-value is technically $P({\rm observation}|H_{0})$, which, as pointed out by the paper, generally does not tell us anything about $P(H_{0}|{\rm observation})$, unless we happen to know the marginal distributions, which is rarely the case in "everyday" hypothesis testing. When we obtain a small p-value and "reject the null hypothesis," what exactly is the probabilistic statement that we are making, since we cannot say anything about $P(H_{0}|{\rm observation})$? The second question relates to a particular statement from page 6(652) of the paper:      Since the p-value, or range of p-values indicated by stars, is not set a priori, it is not the long-run probability of making a Type I error but is typically treated as such.   Can anyone help to explain what is meant by this statement?
26581
I have some discrete times of events and I would like to do a test to see if they are likely to have come from a homogeneous Poisson process.   From this pdf, I see:       REMARK 6.3 ( TESTING POISSON ) The above theorem may also be used to test the hypothesis that a given counting process is a Poisson process. This may be done by observing the process for a fixed time $t$. If in this time period we observed n occurrences and if the process is Poisson, then the unordered occurrence times would be independently and uniformly distributed on $(0, t]$. Hence, we may test if the process is Poisson by testing the hypothesis that the n occurrence times come from a uniform $(0, t]$ population. This may be done by standard statistical procedures such as the Kolmogorov-Smirov test.     However I don't quite understand what to do in practice.  Say my times are   [1, 7, 18, 22, 41, 43, 66, 73, 86, 92]   and the time interval I chose was from $1$ to $100$.  How exactly do I do the Kolmogorov-Smirov test in this example?
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Can someone explain me, how the Law of large numbers and the Gambler's Fallacy do not contradict.  The  says, that there is no memory in randomness and any sequence of events has the same probability as any other sequence.  However, the  says, that given enough repetitions a certain event will likely happen.  To my understanding, these two kinda contradict each other because one says that you can not predict any random event but the other one says so (given enough repetitions of course).  For example imagine a series of coin tosses where the coin comes up heads a million times. The Gambler's fallacy says that the chance for the next toss to be tails is still 1/2. However the law of large numbers says, that since enough repetitions of tosses have come up heads, the next toss is more likely to be tails. (Which is definitely wrong?)
43640
If 20 independent Bernoulli trials are carried out each with a different probability of success and therefore failure. What is the probability that exactly n of the 20 trials was successful?  Is there a better way of calculating these probabilities rather than simply summing together the combinations of success and failure probabilities?
65713
I was reading an entry level statistics textbook. In the chapter on maximum likelihood estimation of the success proportion in data with binomial distribution, it gave a formula for calculating a confidence interval and then nonchalantly mentioned     Consider its actual coverage probability, that is, the probability that the method produces an interval that captures the true parameter value. This may be quite a bit less than the nominal value.    And goes on with a suggestion to construct an alternative "confidence interval", which presumably contains the actual coverage probability.   I was confronted with the idea of nominal and actual coverage probability for the first time. Making my way through old questions here, I think I got an understanding for it: there are two different concepts we call probability, the first being how probable it is that a not-yet-happened event will produce a given result, and the second is how probable is that an observing agent's guess for the result of an already-happened-event is true. It also seemed that confidence intervals only measure the first type of probability, and that something called "credible intervals" measure the second type of probability. I summarily assumed that confidence intervals are the ones which calculate "nominal coverage probability" and credible intervals are the ones which cover "actual coverage probability".   But maybe I have misinterpreted the book (it is not entirely clear on whether the different calculation methods it offers are for a confidence interval and a credible interval, or for two different types of confidence interval), or the other sources I used to come to my current understanding. Especially a comment which I got on another question,      Confidence intervals for frequentist, credible for Bayesian    made me doubt my conclusions, as the book did not describe a Bayesian method in that chapter.   So please clarify if my understanding is correct, or if I have made a logical error on the way.
111121
I study economics and I am trying to solve this problem - without success, unfortunately.     A machine has a defect rate of 6%.       a) Chosen at random 4 pieces (with replacement) from the production flow, compute the probability that none is defective.   b) In the event that extractions are performed 60, compute the probability that there is at least one defective part.   I have no idea how to answer the first question. What logical processes can I use?  Regarding the second question, I thought of using the binomial random variable with $n = 60$ and $p = 0.06$. Then:  $$\begin{multline} P (X \geq 1) = 1 - P (X &lt; 1) = 1 - P (X = 0) = \\ = 1 - \binom{60}{0}  \cdot (0.06)^ 0 \cdot (0.94)^{60} = 1 - 0.02446 = 0.97558.\end{multline}$$  Can somebody please explain me how to solve this problem? Thank you very much.
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I was working on historical data looking for anomalous patterns that we would not expect to occur at random. I'd like to create a scheme to analyze data and markets to test for statistical significance and consistency over time.  Question : Are there some tools we could use to evaluate the nonrandomness of a database? What do we use usually to say that something is random or not (statistically)? Figure we work with change in some value over time.
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I am six years into a business role and have a bachelor's in physics and applied math/stats. Sean Carroll's (Caltech physicist)  opened me to the idea that Bayesian statistics is one useful way of thinking about anything - inevitably you hold a prior and you should update your credence as additional information becomes available.   Is there a path to training your intuition to thinking this way? Critically, it would require repeated practice with verifiable answers through either a course, or self study that includes many problems and solutions. I do not believe simply reading will do.  Possible resources, having read every related question on this site I could find:   "Probability Theory" by Jaynes. Pro: analytic; intuitive explanation of bayesian statistics. Con: prerequisites; missing problems/solutions. "Doing Bayesian Data Analysis" by Kruschke. Pro: includes problems &amp; solutions; requires only "algebra and rusty calculus". Con: works in R, which I think provides for less intuitive learning than the analytical (I may be wrong).   If it is a multi-year path I need to take, starting elsewhere, I am happy to do so! Ideally, I would avoid the frequentist methods, as I have no use for them.  My goal is not to be a scientist, but to leverage insight into how reality works to go above and beyond the established thinking in business.  Many thanks for any suggestions!
64775
I'm working on some problems from Pitman's Probability.  Let events $A$ and $B$ be independent, with $P(A)=0.1$ and $P(B)=0.3$. Find: $(a)$ Neither of the events occur $(b)$ At least one of the events occur $(c)$ Exactly one of the events occur  Attempt: $(a)$ If neither of the events occur, this is the complement of both events occurring. Hence $1-P(A\cap B)=1-P(A)P(B)=1-\frac{1}{30}=\frac{29}{30}$ $(b)$ At least one of the events can be represented as the union of the two events. Hence $P(A\cup B)=P(A)+P(B)-P(A\cap B)=P(A)+P(B)-P(A)P(B)=\frac{4}{10}-\frac{1}{30}=\frac{11}{30}.$ $(c)$ Not sure what this would be. Maybe $P(A^c\;\cap\;B)+P(A\cap B)$?
93821
PART 1: I was reading the article  Wikipedia. I couldn't understand this line:       "if the two events are causally connected ("event A causes event B"), the causal order is preserved (i.e., "event A precedes event B") in all frames of reference."     Is this an assumption or a consequence of STR? Please explain.       Note: My question consists of 2 parts this is the 2nd part. Below is a genral version of my previous question question:.   PART 2: Let there be three events $A$,$B$ and $C$ s.t: $C$ is the result of Simultaneous occurrence of $A$ and $B$. In other words $C$ occurs iff $A$ and $B$ are simultaneous. Now as we know in STR any two events separated in space are not simultaneous in different frames. So In some frames $C$ will occur and in some $C$ will not occur which will cause paradox. I tried many thought experiments to make such a paradox but i failed. In all the experiments that i thought i could not break the causality even by breaking the simultaneity because everytime the fact: "all signals move slower than light" preserved the causality.    So why causlity remains preserved always? Is it due to the fact that nothing can move faster than light?
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I was wondering if there is a way to tell the probability of something failing (a product) if we have 100,000 products in the field for 1 year and with no failures? What is the probability that one of the next 10,000 products sold fail?