I have a statistical sample where D10 = 8 (10% of population under the value 8), D50 = 11 (median), and D90 = 18 (90% of population under the value 18)  Now, I need to find the best fit, in terms of the values $\mu$ and $\sigma^2$ of a lognormal probability distribution, given these D values.   Any help is highly appreciated
110112
What methods can I use to infer a distribution if I know only three percentiles?  For example, I know that in a certain data set, the fifth percentile is 8,135, the 50th percentile is 11,259, and the 95th percentile is 23,611.  I want to be able to go from any other number to its percentile.  It's not  my data, and those are all the statistics I have.  It's clear that the distribution isn't normal.  The only other information I have is that this data represents government per-capita funding for different school districts.  I know enough about statistics to know that this problem has no definite solution, but not enough to know how to go about finding good guesses.  Would a lognormal distribution be appropriate?  What tools can I use to perform the regression (or do I need to do it myself)?
83455
Assume we have $n$ distinct points $x_1,x_2,...,x_n\in \Bbb{R}$, and observations which are independent random variables $y_1,y_2,...,y_n$ depending on formula: $$ y_i=\theta_0+\theta_1x_i^2+\theta_3e^{x_i}+\varepsilon_i $$ where $\varepsilon_i$ are distributed as $N(0,\sigma^2)$. Derive the maximum likelihood estimator for $\theta_{0},\theta_{1},\theta_{2}$ .  I know how to find the MLE for a known distribution like normal or Poisson, but for this question, I don't how I can find the likelihood function, should I consider it as normal distribution? And I don't know what the mean and variance would be since I can't figure out the distribution and likelihood function. Also, should I consider $\varepsilon_i$ as a useful variable, or just treat it as an error in estimation? Thanks for any help.
999
Assuming I need to find the ML estimator for p, p being the chance of success in a Binomial experiment $Bin(N,p)$, I would expect my density function to be:  $$ f(y) = {{N}\choose{y}} p^y(1-p)^{N-y} $$  And so my likelihood function should be:  $$L(p) = \prod_i^n(f(y_i)) =  \prod_i^n({{N}\choose{y_i}})p^{\sum_i^n y_i} (1-p)^{nN-\sum_i^n yi}$$  However, the last exponent, $nN-\sum yi$ seems to be wrong because when I derive the log likelihood and isolate $p$, I get $p = \frac{\sum_i^ny_i}{nN}$, while it should be $p = \frac{\sum_i^ny_i}{N}$. Indeed, when checking online, I find different exponents:   they have $N-\sum yi$, for no obvious reason to me (last time I checked, $\sum_i^n(N-y_i) = nN-\sum y_i$ !)   as well, but there they start from the likelihood function of a Bernoulli experiment. It makes sense that $\sum_i^n(1-y_i) = n-\sum y_i$, but what is more obscure to me is why they take the likelihood function of a Bernoulli while the problem is clearly about a Binomial. I am aware of the link between the two, but not enough to see why their likelihood functions seem to be substitutable to estimate p, especially since it doesn't give me the same result.
111651
Suppose I have a discrete distribution defined by the vector $\theta_0, \theta_1, ..., \theta_N$ such that category $0$ will be drawn with probability $\theta_0$ and so on. I then discover that some of the values in distribution are so small that they underflow my computer's floating point number representation, so, to compensate, I do all my calculations in log-space. Now I have a log-space vector $log(\theta_0), log(\theta_1), ..., log(\theta_N)$.  Is it possible to sample from the distribution such that the original probabilities hold (category $i$ is drawn with probability $\theta_i$) but without ever leaving log-space? In other words, how do I sample from this distribution without underflows?
110892
Since integration is not my strong suit I need some feedback on this, please:  Let $Y$ be $\mathcal{N}(\mu,\sigma^2)$, the normal distrubution with parameters $\mu$ and $\sigma^2$. I know $\mu$ is the expectation value and $\sigma$ is the variance of $Y$.  I want to calculate the $n$-th central moments of $Y$.  The density function of $Y$ is $$f(x)=\frac{1}{\sigma\sqrt {2\pi}}e^{-\frac{1}{2}\left(\frac{y-\mu}{\sigma}\right)^2}$$  The $n$-th central moment of $Y$ is $$E[(Y-E(Y))^n]$$  The $n$-th moment of $Y$ is $$E(Y^n)=\psi^{(n)}(0)$$ where $\psi$ is the Moment-generating function $$\psi(t)=E(e^{tX})$$  So I started calculating:  $$\begin{align}  E[(Y-E(Y))^n]&amp;=\int_\mathbb{R}\left(f(x)-\int_\mathbb{R}f(x)dx\right)^n\,dx \\ &amp;=\int_\mathbb{R}\sum_{k=0}^n\left[\binom{n}{k}(f(x))^k\left(-\int_\mathbb{R}f(x)dx\right)^{n-k}\right]\,dx \\ &amp;=\sum_{k=0}^n\binom{n}{k}\left(\int_\mathbb{R}\left[(f(x))^k\left(-\int_\mathbb{R}f(x)dx\right)^{n-k}\right]\,dx\right) \\ &amp;=\sum_{k=0}^n\binom{n}{k}\left(\int_\mathbb{R}\left[(f(x))^k\left(-\mu\right)^{n-k}\right]\,dx\right) \\ &amp;=\sum_{k=0}^n\binom{n}{k}\left((-\mu)^{n-k}\int_\mathbb{R}(f(x))^k\,dx\right) \\ &amp;=\sum_{k=0}^n\binom{n}{k}\left((-\mu)^{n-k}E\left(Y^k\right)\right) \\ \end{align}$$  Am I on the right track or completely misguided? If I have made no mistakes so far, I would be glad to get some inspiration because I am stuck here. Thanks!
92131
I am trying to find the distribution of sum of 2 lognormal random variables. I referred the literature available on Cross validated, Stack overflow and few papers before posting this.   I used convolution to find the distribution of sum of 2 lognormal rvs. The approximation works for difference. But, not for sum. I am getting a nasty kink at 0 in both CDF and PDF. I could not recognize the reason. With few tweaks, the shape of the distribution comes right. But, I am not sure what I was doing is right.   Can somebody guide me here?
95815
a very naive question for some of you. I have a population of 80000 data points. Some points in the population are many SD away from the rest of the population and they could be easily identified and visualized assigning them a p-value. I know that excluding those values, the population follow a nice normal distribution. What I ask you today is whether you can suggest me an approach to assigning a pvalue to each data point in the population where the null hypothesis is that the data contribute to the normality.
87869
I have an answer but I'm not sure if I'm doing it right: $f(x)$ is the pdf $f(x)=2e^{-2x}$, then, if the random variable $Y=\ln(X)$, then we simply substitute:  $P(Y \leq y)$=  $P(2e^{-2X} \leq y)$  $P(-4X \leq y)$ since $y=ln(x)$  $P(X \leq - \dfrac {1} {4} y)$  This is the distribution function, right?  Thanks!
116793
I'm a mathematician self-studying statistics and struggling especially with the language.   In the book I'm using, there is the following problem:  A random variable $X$ is given as $\text{Pareto}(\alpha,60)$-distributed with $\alpha&gt;0$. (Of course, you could take any distribution depending on one parameter for the sake of this question.) Then a sample of five values $14$, $21$, $6$, $32$, $2$ is given.   First part: "Using the method of maximum likelihood, find an estimate $\hat{\alpha}$ of $\alpha$ based on [the sample]." This was no problem. The answer is $\hat{\alpha}\approx 4.6931$.  But then: "Give an estimate for the standard error of $\hat{\alpha}$."  What is meant by this? Since $\hat{\alpha}$ is just a fixed real number, I don't see in what way it could have a standard error. Am I to determine the standard deviation of $\text{Pareto}(\hat{\alpha},60)$?  If you think the question is not clear, this information would help me as well.
97350
I was wondering, given two normal distributions with $\sigma_1,\ \mu_1$ and $\sigma_2, \ \mu_2$   how can I calculate the percentage of overlapping regions of two distributions? I suppose this problem has a specific name, are you aware of any particular name describing this problem?  Are you aware of any implementation of this (e.g., Java code)?
18953
Does someone know what the integral of the square of the normal distribution with standard deviation sigma over the range $-\infty$ to $+\infty$ is?  Thanks!
88907
This is for my hw, and if anyone can solve the first part of the question it will be great. Here is the question:  Assume a two-class problem with equal a priori class probabilities and Gaussian class-conditional densities as follows:  $$p(x\mid w_1) = {\cal N}\left(\begin{bmatrix} 0 \\ 0 \end{bmatrix},\begin{bmatrix} a &amp; c \\ c &amp; b \end{bmatrix}\right)\quad\text{and}\quad p(x\mid w_2) =  {\cal N}\left(\begin{bmatrix} d \\ e \end{bmatrix},\begin{bmatrix} 1 &amp; 0 \\ 0 &amp; 1 \end{bmatrix}\right)$$  where $ab-c^2=1$.   Find the equation of the decision boundary between these two classes in terms of the given parameters, after choosing a logarithmic discriminant function. Determine the constraints on the values of a, b, c, d and e, such that the resulting discriminant function results with a linear decision boundary.
470
On the , there is this line:      $p(\mathrm{height}|\mathrm{male}) = 1.5789$ (A probability distribution over 1 is OK. It is the area under the bell curve that is equal to 1.)    How can a value $&gt;1$ be OK? I thought all probability values were expressed in the range $0 \leq p \leq 1$. Furthermore, given that it is possible to have such a value, how is that value obtained in the example shown on the page?
103658
I want to fit lognormal distribution to my data, using python scipy.stats.lognormal.fit. According to the , fit returns shape, loc, scale parameters. But, lognormal distribution normally needs only : mean and standard deviation.  How to interpret the results from scipy fit function? How to get mean and standard deviation?
50305
Let's say I have one point that will be taken randomly from a normal distribution with mean $\mu_1$ and standard deviation $\sigma_1$.  Let's say I have another point that is taken much in the same way from another normal distribution with mean $\mu_2$ and standard deviation $\sigma_2$;.  How can I compute the probability, given $\mu_1$, $\mu_2$, $\sigma_1$, and $\sigma_2$, that my first point will be larger than the second?  I am sort of interested in the reasoning behind an "analytic" answer (or as analytic as you can possibly get with the normal distribution, which isn't that much), but I am more importantly looking for an algorithm of computing this probability, as it will be used in a simulation/model.  Does anyone know where I could get started on reasoning through this?  Note: For actual computation, having a table of values of the % of the curve within a given multiple of the standard deviation is feasible in my situation.
121061
I am trying to determine a metric that quantifies the distance between two continuous lognormal distributions. The data is actually a mixture of two lognormal distributions (I am not sure if this can be called bimodal). The individual distributions have different means but the same standard deviations. I have 20 data sets that can be divided into 4 groups, with each group having a constant standard deviation, but the 5 data sets in each group are made up of two distributions with different means. Some examples:      I am in search of a metric that accounts for the contrast (difference in means) and the standard deviation. I have tried out some parameters like coefficient of variation (of the mixture), Ashman's D etc. I have also tried percentage of overlap. However the problem always is that, at the end when plotted against the results I end up with 4 trend-lines again, corresponding to the 4 different standard deviations. Is there some other way I can quantify this?  Just to make myself clear, I want to quantify the difference between two distributions in a single case (between the red and green PDFs in the above image).
42935
My PDF:     M was estimated and found to be 5.   I need to work out the quartiles for the PDF above. In addition, I need to use different methods of estimation to estimate the parameters. So far I've successfully used the method of moments. I'm finding it difficult to form the likelihood function.  This is the approach I took to work out the quartiles, which I believe is not correct; quartile 1 below:    And this is what I have so far for the likelihood function:    Any help would be truly appreciated! Thank you.