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

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Summation of binomial number of poisson random variables

Z is summation of K random variables that each has Poisson distribution with different means. But, K is a Binomial random with parameters of n and p. I was wondering what is the distribution of Z?
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2answers
29 views

If x has a distribution function $F_x(x)$, what is the distribution function of $y = \exp(x)$?

I'm really struggling to figure out this problem from one of my practice exercises for a probability course. I know that the probability distribution function $f_x(x)$ is related to the cumulative ...
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0answers
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Probability Formula for Posterior With 3 Variables

First post on math.stackexchange; pardon me if this is naive/a repeat. I'm following this document here by Prof. David M. Blei: ...
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1answer
15 views

Distribution of random variables when combined

I need help with this problem: If $X$ and $Y$ are two independent random variables and are both standard normal, what is the distribution of $\frac{1}{2}(X^2+Y^2)$? I think I start with ...
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13 views

How do I prove that a given probability distribution is Gaussian

I am trying to plot the distribution of a random variable $x$. I got this distribution by marginalising a wishart distribution. When I plot the distribution curve of $x$, it looks like bell shaped ...
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1answer
21 views

Find conditional probability of random variables

I need to find conditional probability to count mutual information. Random variable X has uniform distribution on set ...
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16 views

Convergence of Uniformly Distributed Random Variables (n-dimensional)

Suppose that ${U_n} = ({U_{n1}},{U_{n2}},...,{U_{nn}})$ is uniformly distributed over the n-dimensional cube ${C_n}={[0,2]^n}$ for each $n=1,2,...$ That is, that the distribution of ${U_n}$ is ...
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0answers
6 views

Unknown bounded continuous distribution

Has the continuous distribution with the following probability density function in $(0,1)$ a name? $f(x;\alpha,\beta)=\frac{1}{\alpha^\beta\Gamma(\beta)}(-\log ...
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4 views

Conditional Probability in Multivariate Normal

Given a tri-variate Normal, the conditional probability of an element given others truncated information is Now if I know that the mean vector u is (-0.91,-1.31,-1.39) and R is ...
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16 views

sum/product combination of random variables

Let $X$ and $Y$ be independent random variables. If I am asked about the distribution of random variable $XY+Y$, is it ok if I compute $XY$ first and then add the result to $Y$ (via convolution, or ...
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18 views

Probability density function definition

The definition above is given in my lecture notes. However there is no further reference/explanation given for what $o(h)$ represents. Can anyone explain this in this case?
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11 views

Approximate a function with a gaussian distribution.

I have a function which has a bell-type graph and i need to find a Gaussian(Normal) with the appropriate mean, variance and constant factor which is close to the original function.The function in ...
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2answers
29 views

Does strictly positive density function on the real line with infinite expected value exist?

The problem is as stated in the title. I am looking for an example or a disproof, whether there exists a continuous density function on the whole real line with infinite expected value. Once again: ...
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15 views

Numerically stable way to compute the conditional covariance matrix

The Wikipedia article on multivariate normal distribution contains the well-known fact about the conditional "sub-distribution": If $μ$ and $Σ$ are partitioned as follows: $$ \boldsymbol\mu = ...
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0answers
15 views

Does martingale model work for betting football matches?

Imagine I have 1 million USD and will be betting 1.000 USD on the win of FC Barcelona each time they play a match in La Liga (Spanish Tier 1 football league). If FC Barcelona loses or ties their last ...
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1answer
30 views

CDF of minimum of correlated and iid random variables

Consider two random variables $X_1=\min (W_1, W_2)$ and $ X_2=\min (W_3, W_4),$ where $W_1$, $W_2$,$W_3$ and $W_4$ are positive, identically distributed random variables. While $W_1$, $W_2$ are ...
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Determine the distribution of the random variable [on hold]

The number of chimney fires in a large city over a week with an average of about 520 fires annually being blames on fireplaces, chimneys or chimney connectors.
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16 views

discrete values probabilities problem [on hold]

Can you help me with this?? An engineer is requested to design water supply and waste water removal systems in a new industrial park consisting of 5 independent buildings. Assume that the water ...
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1answer
21 views

Moment generating function gives an undefined first moment, but first moment still exists?

Let's say we have a probability density function given by $f_X(x) = 2x$ for $0 \leq x \leq 1$. (Note $\int_0^1 f_X(x) = 1$.) The moment generating function is $$\int_0^1 e^{tx}\cdot2x \,dx$$ ...
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log-concavity with PDF and CDF

Assume the following: pdf: $f_X(x)$ cdf: $F_X(x)=P(X \leq x)$ $X$ is a random variable with log-concave pdf $f_X(x)$. $Y = h(X)$ $X \in R^n$ $h: R^n \rightarrow R$ Through the ...
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1answer
41 views

A Question on CDFs and PDFs (substitution/inverse?)

(a) So there has been an answer to the question. Can someone explain how the limits of integration were found? I don't know why the upper limit is going to $X$.
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9 views

sum of two Gaussian random variables conditioned on their sum

I have two independent standard normal R.V.s X and Y, and their sum is Z = X + Y. I am trying to calculate the PDF of X conditioned on Z taking the value z. I know that this is the joint PDF of X and ...
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1answer
8 views

Expectation and variance of X − Y

Let's say I have $X=\min\{X_1,...,X_{10}\}$ with the $X_i\sim Exp(\lambda_i)$ independent. And let $Y=\min\{X_{11},...,X_{20}\}$ What is the expectation and variance of $X-Y$? I really don't know ...
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Given the sum of four Exp(1) distributed random variables, what is the conditional density of sum two of them?

Let T := X+Y+Z+K be indepedent and Exp($1$)- distributed random variables. What is the density of (X+Y) given {T = $1$} ? For M:= X+Y and N := Z+K given {M + N = $1$} The joint density is $ ...
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1answer
27 views

Various modes of convergence of random variables

Let $\lbrace X_n \rbrace_n$ be a sequence of independent random variables such that $$P(\{X_n = \pm 1 \}) = \frac{1}{n}$$ $$P(\{X_n = 0 \}) = 1 - \frac{2}{n}$$ Is the sequence convergent: $1$) almost ...
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16 views

Mean Preserving PDF Spreading

I have a univariate discrete random variable and a histogram representing its PDF (which is asymmetrical). Is there a known way to increase/decrease the variance of the distribution (i.e. scaling it ...
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1answer
29 views

Given the distribution of $X$ and $Y=-2\theta \ln X$. How is $Y$ distributed?

The pdf of $X$ is $f(x) = \theta x^{\theta-1},\enspace 0<x<1, \enspace 0<\theta<\infty.$ Let $Y=-2\theta \ln X.$ How is $Y$ distributed? My work: $$ \begin{align*} F(Y) = P(Y \leq y) ...
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2answers
99 views

Who first found the value of $\int_{-\infty}^{+\infty}e^{-x^2}dx$?

A fairly pretty technique of showing that $$\int_{-\infty}^{+\infty}e^{-x^2}dx = \sqrt{\pi}$$ is to square the integral, writing that square as the product of two integrals with integration variables ...
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1answer
28 views

Expression defined by exponential random variables, probability of being nonnegative

Consider $n \geq 2$. Let $E_1,...,E_n,F_1,...,F_n$ be independent exponentially distributed random variables with rate $1$. Define $T_E = \displaystyle \sum_{i=1}^{n}{E_i}$, and $T_F = \displaystyle ...
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2answers
30 views

Calculating $E[X|Y]$ for continuous $X$ and discrete $Y$

I'm struggling with the following exercise, which I have the solution to but don't understand. I would appreciate any help. The exercise Let $X$ a random variable with $f_X(x) = 2x$ if $x \in [0,1]$ ...
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40 views

A student only wrote down …

In the statistics lecture $6$ discrete and $5$ continuous distributions were discussed. For each distribution one can ask for $\mathbb{P}(X = a), \mathbb{P}(X \leq a), \mathbb{p}(X \geq a), ...
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Find the number of items in $10000$ sets of 10 throws each in which you would expect no even numbers.

Given to us is that we have an irregular six-faced die and the expectation that in $10$ throws, $5$ even numbers show up is twice the expectation that $4$ even numbers show up. The question( as in the ...
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0answers
21 views

Central limit theorem with Lyapunov condition

$Z_1, Z_2,...$ are iid uniformly distributed on $[-1;1]$, $\lim_{n \to \infty} a_n=0$ and $\lim_{n \to \infty} na_n=\infty$ also $a_n>0$ $\forall n$, $X_{n,j}= \frac{1}{a_n}I(|Z_j| \le a_n)$ ...
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1answer
19 views

Computation of two-sided probability density functions from their cumulants using Laplace transform

The computation of one-sided probability density functions (PDFs) from their cumulants using Laplace transform is proposed by following paper: M.N. Berberan-Santos, Journal of Mathematical Chemistry, ...
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1answer
15 views

Is it possible to evaluate a normalizing constant for a characteristic function

Let $X$ be a random variable with density $f$ and characteristic function $\varphi$. Say we know $\varphi$ up to a constant $c$. Is it possible to evaluate this constant using $\int f(x)dx=1$ (or by ...
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21 views

Conditional probabilities given the evidence(Bayesian network)

Let's say we have a Bayesian network: How can I compute P(A | F, E) ? I have all the probabilities for each node. Thanks!
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2answers
52 views

How to compute the sum of geometric distribution [on hold]

How to compute the sum of random variables of geometric distribution $X_{i}(i=0,1,2..n)$ is the independent random variables of geometric distribution, that is, $P(X_{i}=x)=p(1-p)^{x}$, then how to ...
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1answer
14 views

Simplifying this summation

I've been doing this question and I'm stuck! Each customer who enters Larry’s clothing store will, independently of every other customer, purchase a suit with probability p. Assume that N, the ...
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23 views

How to “reduce” a probability distribution satisfying certain conditions

I will try and explain the question I have in term of an example. I am given some probability distribution $f$, in this case of 2 variables x and p, $f(x,p)$. For example, I can pick the ...
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2answers
13 views

metrics for density-sampling similarity, beyond likelihood

I am looking for a metric that would evaluate the distance between a sample $S$ and a density function $D$ Building a sample from a known distribution can be done using a monte-carlo sampling, ...
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2answers
26 views

Convolution of 2 uniform random variables

I really do not know how to do this. Let $X$ have a uniform distribution on $(0,2)$ and let $Y$ be independent of $X$ with a uniform distribution over $(0,3)$. Determine the cumulative distribution ...
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0answers
11 views

normalization of probability density on a surface

Let $p(x)$ ($x\in \mathbb{R}^d$) be a probability density function defined on $\mathbb{R}^d$. Assume we have a closed surface $S\subset \mathbb{R}^d$, on which points follow this function $g(x) = 0$. ...
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0answers
13 views

Can particular outcomes from continuous random variables be said to be probable/improbable?

Suppose a continuous variable $x$ is randomly distributed. For concreteness, let us say that it is Gaussian distributed, $x\sim N(0,1)$, such that $p(x)=\textrm{Gauss}(x; 0, 1)$. From $x$, I can ...
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The expected number of mutations in a sequence of elements, each with random delays

In a sequence, the number of the permutations, is the (minimum) number of the pair of elements needed to switch to make them sorted. For example in the following: ...
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18 views

Convergence in distribution plus convergence of moments.

Suppose that the sequence of r.v $\{X_n\}_{n\geq 1}$ has all the moments, and $X_n\stackrel{D}{\longrightarrow} X\sim N(0,\sigma)$. Assume that $E\{(X_n)^K\} \stackrel{n} {\longrightarrow} E(X^K)$, ...
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28 views

Estimate probability density function of being in a certain time interval

​You arrive at a bus stop in an unfamiliar part of town. Assume that buses arrive at the stop with an unknown (to you) distribution and wait in the bus stop for a few ​minutes. The wait time ...
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17 views

Comparing infinite dimensional distributions

Given two infinite sequences of rvs $(X_{1},X_{2},...)$ and $(Y_{1},Y_{2},...)$, how can we show $(X_{1},X_{2},...)\stackrel{d}{=}(Y_{1},Y_{2},...)$? The way I heard is by comparing all their finite ...
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19 views

ML estimation for Weibull

What are the maximum likelihood estimators of $\eta$ and $\beta$ ($\eta>0$ and $\beta>0$) for an i.i.d. sample of size $n$ from the following density: $f(y_i)=\frac{\beta x_i^{\beta-1} }{\eta ^ ...
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1answer
27 views

Calculate probability of joint PDF

I'm given the following joint PDF and asked to calculate $P(X+Y>1)$ $f_X$$_Y$$(x,y)=2/5$ for $0<y<1$ & $0<x<5y$ and $f_X$$_Y$$(x,y)=$ $0$ else I know I have to take the ...
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54 views

Using Jensen's inequality to prove the Cauchy distribution has no mean

I can see that there is no mean because $\int x / \pi(1+x^{2})$ does not converge from -inf to inf. But my prof hinted at using Jensen's inequality for the proof. $$f(E(X)) \le E(f(X))$$ How can I ...