If I generate a Normal and leptokurtotic continuous probability distribution (pdf), how can I measure the mass of these pdf's? If both pdf's have the same number of sample observations, but obviously have different statistical moments, can we expect that the area/mass that these two distributions enclose be different or identical to one another? How does the concept of pdf mass compare to the concepts and estimation of pdf moments and shape?
123293
If we minimize the Wasserstein loss, $$W_1 (P_S, P_T) = \underset{\gamma \in \Pi}{\text{min }} \sum_{x,y} |x-y|\gamma(x,y)$$ which means we are looking for the coupling $\gamma(x,y)$ that minimizes the cost of moving mass from the source distribution $P_S$ to the target distribution $P_T$,  Do the masses of the source and target distributions have to be identical? If so or not, how are the masses of $P_T$ and $P_S$ accounted for mathematically in, or when minimizing, the shown formula, or what might be missing/overlooked by the Wasserstein approach overall due to the mass differences?
13236
I want to understand how to prove results regarding the relationship between the standard and nonstandard normal distributions. In other words, I want to prove the results regarding how to use z tables with nonstandard normal distributions, and finding its p'th percentiles.  What books or articles should I get through to do that? I have taken courses on calculus, real analysis and complex analysis. Do I need anything else to prove the results?
6196
I would like to learn how to calculate the expected value of a continuous random variable. It appears that the expected value is $$E[X] = \int_{-\infty}^{\infty} xf(x)\mathrm{d}x$$ where $f(x)$ is the probability density function of $X$.  Suppose the probability density function of $X$ is $$f(x) = \frac{1}{\sqrt{2\pi}}e^{\frac{-x^{2}}{2}}$$ which is the density of the standard normal distribution.  So, I would first plug in the PDF and get  $$E[X] = \int_{-\infty}^{\infty} x\frac{1}{\sqrt{2\pi}}e^{\frac{-x^{2}}{2}}\mathrm{d}x$$ which is a rather messy looking equation. The constant $\displaystyle\frac{1}{\sqrt{2\pi}}$ can be moved outside the integral, giving  $$E[X] = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} xe^{\frac{-x^{2}}{2}}\mathrm{d}x.$$  I get stuck here. How do I calculate integral? Am I doing this correctly this far? Is the simplest way to get the expected value?
49358
It's a little tricky to me, I am not sure how to proceed...my atempt was to stick strictly to the definitions given and the formula for the (continuous) expected value.  Prove that $E[e^{\mu+\sigma Z}\mathbb{1}_{Z&gt;-d}]=e^{\mu+\frac{\sigma ^2}{2}}\Phi(d+\sigma)$, where $\Phi$ is the standard normal distribution function, $Z  \sim N(0,1)$(i.e. standard normal distribution) and $\mathbb{1}_{Z&gt;-d}$ denotes the following indicator function of an event such that, $$\mathbb{1}_{Z&gt;-d}=1 \text{ when } Z &gt;-d \text{ and, } \mathbb{1}_{Z&gt;-d}=0 \text{  otherwise}$$ during your work, denote the standard normal density function as $\phi$.  I only got up to thinking $e^{\mu+\sigma Z}\mathbb{1}_{Z&gt;-d}$ is essentially just $e^{\mu+\sigma Z}$ if $Z&gt;-d$ for some $d$ and $0$ otherwise. And so, the expected value is defined (for continuous random variables) $E[X]= \int_{-\infty}^{\infty}xf(x)dx$ where $f(x)$ is the probability for each random variable $x$. I thought this would lead me to using $\phi$ and ultimately leading me to $\Phi$ in my final conclusion as in the RHS. But I am not sure about how to proceed, would someone please help me? It would be great if you could explain the steps to me...thank you very much in advance...
97211
In my statistics class, we're just beginning to talk about (point) estimation. I have a small question that might actually be due more to notation/definition than anything very conceptual:  Say you have a population $P$ of students in a classroom and you want to find the mean height by randomly selecting $n$ students $s_1,...,s_n$ in $S\subseteq P$ to measure. My book calls $X_1,...,X_n$ a random sample. I'm just a little confused as to what $X_i$ is defined to be (what $X_i$, as a random variable, is mapping from). In this example, my book would say {$X_1,...,X_n$} are the respective heights of the $n$ randomly selected students. Just to make sure I have the right picture in my head: $S$ is the ("first") sample space and $X:S\to\mathbb{R}$ is a random variable ($X$ is the "height" function defined on $S$). We denote $X_i$ as the restriction of $X$ to the $s_i$-th student in $S$. That is, we define $X_i:${$s_i$}$\to\mathbb{R}$ so $X_i(s_j)$ is not defined for $i\neq j$, and:   $X_1(s_1)=X(s_1)=x_1$  $X_2(s_2)=X(s_2)=x_2$  $.$  $.$  $.$  $X_n(s_n)=X(s_n)=x_n,$ where $x_1,...,x_n$ are the respective heights of the $n$ students. Then $Range(X)=${$X_1(s_1),...,X_n(s_n)$}$=${$X_1,....,X_n$} is considered the new sample space that we define the pmf/pdf $f$ on, where $f:\mathbb{R}\to\mathbb{R}$ and $f\circ X(s_1)=f(x_i)$.   Is this the right idea?
3016
This is an elementary question, but I wasn't able to find the answer.  I have two measurements: n1 events in time t1 and n2 events in time t2, both produced (say) by Poisson processes with possibly-different lambda values.  This is actually from a news article, which essentially claims that since $n_1/t_1\neq n_2/t_2$ that the two are different, but I'm not sure that the claim is valid.  Suppose that the time periods were not chosen maliciously (to maximize the events in one or the other).  Can I just do a t-test, or would that not be appropriate?  The number of events is too small for me to comfortably call the distributions approximately normal.
98634
How do I generate $n$ identically distributed but not independent normal random numbers such that their sum falls within a prespecified interval $[a,b]$ with probability $p$?  (This question is motivated by generating a random walk that ends up at a prespecified point: . Since a continuous random variable has zero probability of reaching an exact number, we do the second best thing and ask for an entire interval to end up in.)    EDIT:  has been proposed as a duplicate, which in turn is closed as a duplicate of . I agree that both of these are helpful. However, the point of the current question (more specifically, of the answer) is to first figure out that we can use a multivariate normal distribution to address the question, and second, what kind of covariance matrix works. How to sample from a distribution with that covariance is a third step, where the linked threads are helpful.
87585
I know that if the moment generating function of two distribution converges to the same function then the two distribution converges in CDF. But how can we prove this thing explicitly ?
13641
I am a grad student developing an interest for statistics. I like the material over-all, but I sometimes have a hard time thinking about applications to real life. Specifically, my question is about commonly used statistical distributions (normal - beta- gamma etc.). I guess for some cases I get the particular properties that make the distribution quite nice - memoryless property of exponential for example. But for many other cases, I don't have an intuition about both the importance and application areas of the common distributions that we see in textbooks.   There are probably a lot of good sources addressing my concerns, I would be glad if you could share those. I would be a lot more motivated into the material if I could associate it with real-life examples.
62388
How can we derive the chi-squared probability density function (pdf) using the pdf of normal distribution?  I mean, I need to show that   $$f(x)=\frac{1}{2^{r/2}\Gamma(r/2)}x^{r/2-1}e^{-x/2} \&gt;, \qquad x &gt; 0\&gt;.$$
24200
Suppose we have $v$ and $u$, both are independent and exponentially distributed random variables with parameters $\mu$ and $\lambda$, respectively.   How can we calculate the pdf of $v-u$?
99479
We are typically introduced to method of moments estimators by "equating population moments to their sample counterpart" until we have estimated all of the population's parameters; so that, in the case of a normal distribution, we would only need the first and second moments because they fully describe this distribution.   $E(X) = \mu \implies \sum_{i=1}^n X_i/n = \bar{X}$  $E(X^2) = \mu^2 + \sigma^2 \implies \sum_{i=1}^n X_i^2/n$  And we could theoretically compute up to $n$ additional moments as:  $E(X^r) \implies \sum_{i=1}^nX_i^r /n$  How can I build intuition for what moments really are? I know they exist as a concept in physics and in mathematics, but I find neither directly applicable, especially because I don't know how to make the abstraction from the mass concept to a data point. The term seems to be used in a specific way in statistics, which differs from usage in other disciplines.  What characteristic of my data determines how many ($r$) moments there are overall?
33721
Supppse $X$ and $Y$ are standard uniformly distributed in $[0, 1]$, and they are independent, what is the PDF of $Z = Y / X$?   The answer from some probability theory textbook is  $$ f_Z(z) =  \begin{cases} 1/2, &amp; \text{if } 0 \le z \le 1 \\ 1/(2z^2), &amp; \text{if } z &gt; 1 \\  0, &amp; \text{otherwise}. \end{cases} $$  I am wondering, by symmetry, shouldn't $f_Z(1/2) = f_Z(2)$? This is not the case according to the PDF above.
15769
The  is a discrete probability distribution over a set $\{1,\dots, N\}$ with the probability mass function  $$ p(1)=\frac{1}{N},\qquad p(k)=\frac{1}{k(k-1)}\quad\text{for }k\in\{2,\dots, N\} $$  I'd like to use it as part of an implementation of an , ideally in Python where a uniform random number generator is available.
29524
The kurtosis is to measure the peakedness and flatness of a distribution. The density function of the distribution, if it exists, can be viewed as a curve, and has geometric features (such as curvature, convexity, ...) related  to its shape.  So I wonder whether the kurtosis of a distribution is related to some geometric features of the density function, which can explain the geometric meaning of kurtosis?
39340
It seems that a number of the statistical packages that I use wrap these two concepts together. However, I'm wondering if there are different assumptions or data 'formalities' that must be true to use one over the other. A real example would be incredibly useful.
7680
Say I have two normal distributions A and B with means $\mu_A$ and $\mu_B$ and variances $\sigma_A$ and $\sigma_B$. I want to take a weighted mixture of these two distributions using weights $p$ and $q$ where $0\le p \le 1$ and $q = 1-p$. I know that the mean of this mixture would be $\mu_{AB} = (p\times\mu_A) + (q\times\mu_B)$.  What would the variance be?    A concrete example would be if I knew the parameters for the distribution of male and female height. If I had a room of people that was 60% male, I could produce the expected mean height for the whole room, but what about the variance?