# Tagged Questions

Questions on using, finding, or otherwise relating to probability distributions, pdfs, cdfs, or the like.

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### Why Characteristic function is given such a name?

I've come to know that, For random variable $X$ ,with a Probability mass function P, $\phi_X(t)$ defined by : $\phi_X(t) : \mathbb R \to \mathbb C$ $\phi_X(t) = E(e^{itX}) =E[\cos tX + i\sin tX]$ ...
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### Calculating the probability distribution function of a realization of a random process

I have a realization of a continuous random process, $y = f(t)$, a function of time ($t$). I am trying to calculate $P(y = Y_0)$, the probability distribution function of $f(t)$. Am I right in saying ...
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### Transforming a categorical distribution by repeating trials and taking a plurality

Suppose you have a K-sided, weighted die. This is represented by a categorical distribution. Now, let's say you roll the die N times, and then pick a "winner" by choosing whichever outcome has a ...
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### Need help finding joint distribution of uniform and exponential

Let X be an exponential random variable with parameter λ and Y be a uniform random variable on [0,1] independent of X. Find the probability density function of X + Y. Now I have computed this ...
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### How we can show $\mathbb{E}[T]=0$ and $\operatorname{Var}(T)=\frac{n}{n-2}$.

I need help with this question. Let $Z\sim N(0,1)$ and $Y\sim X^2_{(n)}$ be independent variables, and define$$T\stackrel{\rm def}{=}\frac{Z}{\sqrt{\frac{Y}{n}}}.$$ Prove that $\mathbb{E}[T]=0$...
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### Inequality for binomial distribution

In a bombing attack there is $50\%$ chance that any bomb will hit the target. Two direct hits are required to destroy the target completely. How many bombs must be dropped to give a $99\%$ chance or ...
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### Bivariate normal concave downwards

The Hessian matrix of the bivariate standard normal distribution (where both standard normal distributions are independent) is positive definite. Yet, it concaves downwards. If the Hessian is positive ...
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### Maximum-a-posteriori estimation with Gamma prior and scale-invariant likelihood

Let $\mathbf{X}$ be a vector of parameters with prior distribution $X_i \sim \text{Gamma}(\alpha, \beta)$ i.i.d. for $i = 1, \ldots, n$. Let us denote this prior by $p(\mathbf{X})$. We get to observe ...
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### expectation of discrete uniform distribution (general case)

You may be thinking - can't he find this answer somewhere else? Well I've tried but my textbook is very concise and questions so far have been very problem specific. Well I actually have the answer ...
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### What is the distribution of the square of a sum of correlated Gaussian random variables?

Suppose a random vector $X\in\mathbb{R}^n$ follows a centered multivariate Gaussian distribution with zero mean and covariance $\Sigma$. We know that a linear combination of every elements in $X$ ...
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### Proof Attempt: Non-decreasing continuous CDF is standard uniformly distributed

Proof Attempt: For any random variable $X$ with non-decreasing continuous cdf $F(x)=\Pr(X≤x)$ (note that the inverse function does not necessarily exist due to flat regions in $F$), I wish to prove ...
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### Inferring absolute continuity of summands from absolute continuity of sum

Suppose we have i.i.d. random variables $X_1, X_2$. If $\text{Law}(X_1 + X_2)$ is absolutely continuous with respect to the Lebesgue measure $\lambda$, can we infer that each $X_i$ is absolutely ...
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### How to find (GEV) distribution parameters with optimization?

I'm currently trying to replicate this study with python. http://pages.stern.nyu.edu/~sfiglews/Docs/RND_draft7.pdf The section I'm currently working on is between p.17-20 in the study. The study ...
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### Comparing two samples with weights

Given two sets of samples and their respective weights, what is a sensible way sensible way of comparing these two, i.e., if they are representing the same distribution? Some things that came into my ...
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### Confusion about the average distance traveled on a $1$D random walk

The average absolute distance on a one dimensional random walk is supposed to be $\sqrt{n}$. Where $n$ steps are taken from the origin or $n$ is the time. I don't have an intuitive understanding or ...
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### Find the relation between mean and variance for lognormal distribution giving as input mean and standard deviation of normal distribution

I am working with a Lognormal distribution with mean $m$ and variance $v$. I give as input $\mu$ and $\sigma$ of the relative normal distribution in order to calculate the cumulative Lognormal. Now, ...
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### Intuition of the relation between poisson process and order statistics

Lemma: Let $T_n$ be the time of the nth arrival in a Poisson process and $U_k$, $k=1,2....n$ be independent uniform on $(0,1)$. Then the order statistics of $U_1, U_2,....,U_n$ have the same ...
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### Find the Expectation of $2\sum\limits_{i=1}^n \frac {(y_i-\alpha)^2-\beta ^2}{(\beta^2+(y_i-\alpha)^2)^2}$ for $y_i$ i.i.d. Cauchy$(\alpha,\beta)$

Find the Expectation of $$2\sum_{i=1}^n \frac {(y_i-\alpha)^2-\beta ^2}{(\beta^2+(y_i-\alpha)^2)^2}$$ given $y_1,y_2...$ iid ~Cauchy$(\alpha,\beta)$ with pdf $(-\infty < y<\infty , \beta>0)$: ...
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### The problem of ksdensity plot in Matlab

I want to verify that random numbers generated by exprnd match the exponential distribution. I used Matlab's ksdensity function –...
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### how far the distribution from the uniform distribution

I have two discrete probability distributions $P$ and $Q$, where $P=(p_1,...,p_n)$ and $Q=(q_1,...,q_n)$, in addition I have uniform distribution $U=(\frac{1}{n},...,\frac{1}{n})$. The question is ...
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### Computing this conditional expectation?

Consider two random variables $X_1$,$X_2$ iid from a cdf $G$ on $[0,b]$. Consider also some concave function $f(.)$ with inverse $\phi(.)$ over $[0,b]$ such that $f(x)\leq x$, $\forall x\in [0,b]$. ...
Suppose a continuous random variable has odd moments = zero and even moments as follows $$E[X^{2n}] = \frac{(2n)!}{2^n n!}$$ Then the mgf is, by Maclaurin series expansion where we can switch sum ...