# Questions tagged [probability-distributions]

Questions on using, finding, or otherwise relating to probability distributions, probability density functions (pdfs), cumulative distribution functions (cdfs), or other related functions.

17,539 questions
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### Statistics and Probability Expected Value Questions

Compute the frst two moments (that is EY and EY^2) of Y = 2^-X if X = b(n, p) (binominal distribudion). Random variable X has the distribution P(X = k) = c*3^-k(keN). Determine the value of c. Does ...
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### Inequalities for standardized central moments of probability distributions

It's known that the standardized central even moments of any probability distribution with a density symmetric around the mean form a non-decreasing series, the lower bound (when all are equal to 1) ...
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### How to compute the optimal values of $\lambda_1,\ldots,\lambda_K$?

So, here's a question that has come up in my research work. Suppose $P_1$ and $P_2$ are $M\times M$ transition probability matrices such that $P_1\neq P_2$. Further, let $\mu_1$ and $\mu_2$ denote the ...
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### Distribution of $\frac{X^{t}AX}{X^{t}X}$ in multivariate normal distribution

In the multivariate normal case for $X\sim N_{p}(0,I)$, consider an idempotent matrix $A$ of rank $k<p$. I need to find the distribution of $\frac{X^{t}AX}{X^{t}X}$. I know that $X^{t}AX$ follows ...
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### Express $P\{\{X_1+X_2\leq \alpha \}\cap\{X_3\leq X_2\}\}$ using CDF and PDF

Let $X_1$ $X_2$ and $X_3$ be three positive independent random variables. The PDF and CDF of $X_i$ are denoted by $f_{X_i}(x_i)$ and $F_{X_i}(x_i)$ respectively. I would like to compute the ...
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### Maximum Likelihood Estimate with Multiple Parameters

I am not very familiar with multivariable calculus, but something tells me that I don't need to be in order to solve this problem; take a look: Suppose that $X_1,...,X_m$ and $Y_1,...,Y_n$ are ...
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### What sampling distribution does $\hat{p}$ in this scenario follow?

Homework question, not sure whether I should be using a normal distribution, a binomial distribution or a hypergeometric distribution. A mail-order company promises its customers that the products ...
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### the probability distribution of Y is P{Y=-1}=p P{Y=1}=1-p,(0<p<1),set Z=XY. [on hold]

Let the random variables $X$ and $Y$ be independent,$X$ follows the exponential distribution with parameter $1$, and the probability distribution of $Y$ is $P${Y=$-1$}=$p$ $P${Y=$1$}=$1$-$p$,set$Z=XY$...
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### Incoming call in a business processing unit… [on hold]

a) Incoming calls are monitored in a Business processing unit. Call arrival is modeled as Poisson distribution with an average rate of $18$ per hour. Determine the probability that the time interval ...
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### Overlap lower bound between distributions

Suppose that I have multinomial distribution with probabilities $p_i$, for $i \in [1,n]$. I'd like to find a lower bound for the sum of the overlapping probabilities between $p_i$'s and the ...
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### Poisson-Approximation of Poisson-Binomial distribution

I'm looking for a proof as to why the Poisson-Binomial distribution can be approximated by the Poisson distribution. A Poisson-Binomial distribution is a sum of independent Bernoulli trials that are ...
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### Posterior of a corrupted transformation

Let $z\sim\mathcal{N}(z|0,1)$ and let $p(x|z)=\mathcal{N}(x|f(z),\gamma)$ where $f:\mathbb{R}\to\mathbb{R}$ is a differentiable bijection. I am trying to approximate the posterior $p(z|x)$ with a ...
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### Intuitive explanation of CDF of a Binomial distribution in the volume of a Hyperspherical Cap

Note: This is my first question ever in stackexchange, I apologize for any mistakes in formatting, on the appropriateness of the question and tags. From Wikipedia, I know the regularized incomplete ...
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### What is considered to be a Heap’s law?

I’m not sure if this is more physics question than mathematics but anyways. Something is usually said to follow Heap’s law if it is given as a function $V(n)=K n^b$, where $b$ and $K$ are constants (...
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### Is $\Pr[X \leq x] \leq \max(\Pr[X + Y \leq x], \Pr[X - Y\leq x])$

Suppose that $X$ and $Y$ are random variables which can be arbitrarily dependent. Does the following inequality hold: \Pr[X \leq x] \leq \max(\Pr[X + Y \leq x], \Pr[X - Y\leq x]). \...
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### Uniqueness of 1-dimensional distributions via uniqueness of Laplace Transform

Consider a stochastic process $X = (X_t)_{t \geq 0 }$ on $\mathbb{R}$ and two probability measures $\mathbb{P}_1,\mathbb{P}_2$ with expectations denoted $\mathbb{E}_1,\mathbb{E}_2$. Suppose for $g$ ...