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

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5
votes
2answers
163 views

Does the Poisson distribution work for non-integer exponents?

The question regards the Poisson distribution function as given by: $$\frac{x^k e^{-x}}{k!}$$ The distribution's domain (x) goes from 0 to $\infty$, and $k \in \mathbb{N_0}$ I tried the ...
1
vote
1answer
16 views

Joint PDF to CDF

So I have this joint PDF: $$ f(x,y)= \begin{cases} 4xy & \text{ for } 0 \leq x \leq 1, 0\leq y \leq 1\\ 0 & \text{ otherwise} \end{cases} $$ To make this a CDF, I have tried to double ...
1
vote
0answers
31 views

Linear Probability Density Transformations

Suppose that $\mathbf{y=Ax}$ and that a probability density function over $\mathbf{x}$ is defined as $p(\mathbf{x})$. If $\mathbf{A}$ has an inverse then the PDF over $\mathbf{y}$ is given by ...
1
vote
1answer
34 views

Generate random numbers with a modified PERT distribution

I want to generate random numbers based on the modified PERT distribution. The modified PERT distribution is a special case of the beta distribution and is defined as: $$f_X(x) = ...
0
votes
1answer
19 views

Infimum of Gamma distribution

Let $X$ be a Gamma random variable with the CDF $F_X(x)=\frac{1}{\Gamma(\alpha)}\gamma(\alpha,\beta x)$ where $\Gamma(x)$ represent the gamma function and $\gamma(a,b)$ denotes the lower-incomplete ...
0
votes
1answer
36 views

How to prove that convergence in MGF implies Convergence in Distribution?

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 ?
1
vote
1answer
50 views

Extreme Value Theory - Show: Normal to Gumbel

The Maximum of $X_1,\dots,X_n. \sim$ i.i.d. Standardnormals converges to the Standard Gumbel Distribution according to Extreme Value Theory. How can we show that? We have $$P(\max X_i \leq x) = ...
0
votes
1answer
34 views

Distribution of marbles on number line

I have a set of marbles and a number line from 0 to infinity. Every step I either put a new marble on the number 0 or I move one existing marble (chosen uniformly) to the next number. The ratio ...
0
votes
0answers
43 views

Total Integral of ordered joint probability is 1 ???

I have four random variables $X_1$, $X_2$, $X_3$ and $X_4$. Their joint dist. is $f(x_1,x_2,x_3,x_4)= \exp(-x_1-x_3)$, where limits are $x_4 = 0$ to $\infty$, $x_3 = x_4$ to $\infty$, $x_2 = x_3-x_4$ ...
0
votes
1answer
30 views

A moment's question.

Let G be a (absolutely) continuous distribution such that $$\displaystyle{\int_{-\infty}^{\infty}{x^{2}dG(x)}}<\infty$$ or $$\displaystyle{\int_{0}^{1}{\left[G^{-}(t)\right]^{2}dt}}<\infty.$$ ...
22
votes
1answer
393 views

Zombie outbreak on a $k$-regular graph

Suppose we have a zombie outbreak on a connected $k$-regular graph of order $n$. There are $n_0$ initially infected zombie nodes, and each turn, each zombie infects its neighbors with probability ...
0
votes
0answers
129 views

Is there a way to find a steady state probability density for a given transformation?

I was looking at probability density transformations here. They use $g(y) = f(x(y))|dx/dy|$ where $x(y)$ is the inverse of the transformation. Is there a way to, given only the transformation, find ...
1
vote
1answer
49 views

What's the distribution of $\Phi(X), X \in N(0,1)$?

In a course on statistics, this set of non-compulsory exercises were supplied (in Swedish). I'm stuck on 8.10. My translation of the exercise: The stochastic variable X has a $N(0,1)$ ...
0
votes
0answers
19 views

Repetitive sampling from the uniform and an unknown distribution

I am trying to model an experiment. We have n ''players''. Each one picks independently a sample from a continuous uniform distribution in the [0,$2^{64}$], let's call it $u_i$. He also picks a sample ...
0
votes
0answers
16 views

Distribution maximum with small sample related to large sample

Suppose the random variables $X_i$, $i=1,\cdots,n$ and $Y_j$, $j=1,\cdots,m$ all have distribution $F(x)$, with order statistics denoted by $X_{(i)}$ and $Y_{(j)}$. Assuming $n<m$ (e.g. $n=m/100$), ...
3
votes
0answers
28 views

Gamma distribution Norming constant for extreme minima

the norming constants for extreme maxima of Gamma distribution is known and is give in link.springer.com/article/10.1007/s10687-010-0125-3. I would like to know is there reference or paper that states ...
0
votes
0answers
37 views

Mathematical Probability and Statistics( all the math need)

I would like some suggestions about mathematical techniques and knowledge are required to understand and master 2nd year undergraduate probability and statistics. I am mature student with some ...
1
vote
0answers
29 views

Showing $\lambda_V(x)\leq \min\{\lambda_1(x),\cdots, \lambda_n(x)\}$.

Suppose $X_1, \cdots, X_n$ are independent, nonnegative continuous functions, each $X_i$ has hazard function $\lambda_i(x)$. If $V=\max\{X_1, \cdots, X_n\}$, I need to show that ...
1
vote
1answer
19 views

Independence of distribution

Let there be a random matrix defined as $\mathbf{H}_1 = X + \boldsymbol\nu$, where, $X$ is deterministic and $\boldsymbol\nu$ is Gaussian white noise. Now let there be another random matrix defined as ...
0
votes
2answers
23 views

Expected Value with Parameter p

The random variable X has the following probability distribution: P[X=-1]= (1-p)/2 P[X=0]= 1/2 P[X=1]= p/2 The parameter p satisfies the inequality $0 < p < 1$. Find the expected value and ...
0
votes
1answer
19 views

Negative Binomial distribution as a Gamma mixture distribution

Let $f(x;\theta)$ be the poisson frequency function with mean $\lambda$. and $p(\lambda)$ the Gamma distribution with mean $\mu$, and variance $\mu^2/\alpha$. I have to show that ...
-1
votes
0answers
34 views

A partial derivative wrt a limit of a double integral

I'm stuck with this problem. Is there a way to figure out the sign of the partial derivative $\dfrac{\partial x}{\partial a}$ where $x$ is the solution of the following equation: $$ ...
3
votes
1answer
59 views

What is the joint probability distribution of number of balls after $n$ draws?

The following problem came into my mind when I am studying the Polya Urn Model. At the beginning, from a bin containing $c_1$ balls labeled $1$, $c_2$ balls labeled $2$, … , $c_m$ balls labeled $m$, ...
0
votes
1answer
32 views

Normalizing constants for Extreme value distributions

I have a question regarding the normalizing constants $\mu$ and $\sigma$ that appear in the following problem. Let the random variable $Y_n$ be $Y_n=max(a_1,a_{2},\cdots, a_n)$ and $X_{n}$ be ...
1
vote
1answer
55 views

What distribution is this?

Top: Uniform, Bottom: ?? Distribution. Ignore the random spikes - those are just binning errors. Looking for a distribution that is on $[0,1]$ and is equal to $0$ at $1$ and some positive ...
1
vote
2answers
46 views

If $X$ is distributed normally with mean $0$, is it correct to say $X$ and $-X$ “have the same distribution”?

Q: If $X$ is distributed normally with mean $0$, is it correct to say $X$ and $-X$ have the same distribution? In a way, this seems correct: both $X$ and $-X$ have the same probability density ...
0
votes
1answer
26 views

Using the Weibull Distribution, derive $E(X^k)$

If $X$~WEI$(\theta,\beta)$, derive $E(X^k)$ assuming $k\gt-\beta$. Note that $X$~WEI$(\theta,\beta)=\frac{\beta}{\theta^{\beta}}x^{\beta -1}e^{-({x}/{\theta})^{\beta}}$ I am having a very difficult ...
1
vote
0answers
20 views

Deriving joint distribution from expectation

Given two random variables $X$ and $Y$ and let $K$ be a constant value. Assume the expectation $\mathbb{E}[X(Y-K)^{+}]$ is given for all possible values of $K\geq 0$. Is there a way to derive the ...
0
votes
0answers
7 views

What is a residual distribution vector

I have a general question about a probability distribution of the following form. Let us assume $\mu_0$ is a propability distribution on the set $A=\{1,2,\dots,n\}$ (in my case its the starting ...
4
votes
2answers
53 views

Homework problem - Ways to test if a density function is cumulative density function

I have a problem that states: Let $F : \mathbb R \to R$ be defined by $$F(x) =\begin{cases}e^{\frac{-1}{x}} &\text{if } x > 0\\ 0&\text{if } x \leq 0\end{cases}$$ Is $F$ a ...
0
votes
1answer
28 views

Conditional probability and distribution

Let Y ∼ Exp(1/5). Find P(Y ≤ 18|Y > 13). Could anyone give me any hints?
0
votes
1answer
32 views

Joint distribution of independent random variables

Say I have two independent random variables $X$ and $Y$ both having the exponential distribution. I.e. $f_X(x) = \lambda_1 e^{-\lambda_1 x}, \ x \ge 0, 0$ elsewhere $f_Y(y) = \lambda_2 e^{-\lambda_2 ...
-1
votes
0answers
29 views

Homework help - Random Variable min - can't understand what teacher wants me to do with problem

The problem is: Let X(1), . . . ,X n be independent random variables, with X(i) having an exponential with parameter λ(i) distribution, for any i. Then the distribution of the random variable X = ...
1
vote
6answers
169 views

Producing a CDF from a given PDF

So I have this PDF: $$ f(x)= \begin{cases} x + 3 & \text{ for } -3 \leq x < -2\\ 3 - x & \text{ for } 2 \leq x < 3\\ 0 & \text{ otherwise} \end{cases} $$ To make this a CDF, I ...
1
vote
1answer
18 views

Determine the values of c so that the following functions represent joint probability distributions of the random variables X and Y

Determine the values of c so that the following functions represent joint probability distributions of the random variables X and Y: f(x,y) = c x y, for x = 1,2,3; y = 1,2,3; f(x,y) = c|x-y|, for x ...
2
votes
2answers
61 views

Probability Distributions and Probability

Suppose $X \sim N(3, 4)$, and let $Y = X^2$. Find $\Pr(Y ≥ 12)$. What does $Y$ mean?
0
votes
2answers
33 views

Probability of CDF and PDF [closed]

Suppose continuous random variable $X$ has a cumulative distribution function $FX$ satisfying $FX(x) = 2x^2 − x^4$ for $0 \leq x \leq 1$. (a) Compute $\displaystyle P\left(\frac{1}{4}\leq X \leq ...
2
votes
3answers
87 views

distribution of infinite sum of $\sum (2x_n -1)/2^n$

$\{X_n\}\sim\mathrm{Bernoulli}(\frac {1}{2})$ $$Y=\sum_{n=0} ^{\infty} \frac {2X_n -1}{2^n}$$ Find the distribution of $Y$ $X_n$ are independent
0
votes
0answers
26 views

Notation related to Markov kernels

We wish to jointly construct two copies $(X_n)_{n \in \mathbb{N}}$ and $(Y_n)_{n \in \mathbb{N}}$ of a Markov chain on general state space, s.t. for $n=1,2,...$ $\mathcal{L}(X_{n+1}|X_n) = ...
0
votes
2answers
53 views

Homework help finding pdf's of y given pdf's of x - stuck

If anyone can give me the steps on how to find pdf$\,'$s of $y$ given $x$. Let X be a continuous random variable with probability density function given by $$ {\rm f}\left(x\right) ...
0
votes
1answer
15 views

Is squared Brownian Motion a gaussian process?

I am working at the following SP, given by $(X_t)_{t\geq0} = \alpha W_t^2+\beta t$ where $W_t$ is Brownian motion and $\alpha,\beta$ real. I managed to calculate mean and covariance function and now I ...
0
votes
1answer
28 views

Conditioning on independent coin tosses - general solution to brute force method?

Consider 10 independent tosses of a biased coin with a probability of heads, $p$. question (4d): find the probability there are 5 heads in first 8 tosses and 3 heads in last 5 tosses. I managed to ...
0
votes
1answer
29 views

Limiting distribution of $n(T_n-4p^3(1-p))$

I want to find the limiting distribution of a $n(T_n-4p^3(1-p))$, where $T_n=\displaystyle\frac{4(n-t)t(t-1)(t-2)}{n(n-1)(n-2)(n-3)}$ with $t=\sum X_i$ is the UMVUE of $4p^3(1-p)$ that I found, where ...
1
vote
2answers
31 views

When to use alternate parametrization of Gamma distribution?

In Loss Models, 4th ed., by Klugman et al., the following parametrization is given for the Gamma distribution: $$f(x) = \dfrac{(x/\theta)^{\alpha}e^{-x/\theta}}{x\Gamma(\alpha)}\text{.} $$ When ...
0
votes
1answer
32 views

Taking an integration with joint probability integrand.

I encounter a joint probability and I was wondering Am I allowed to rewrite it as an integral by using law of total probability? $$P( \{ f(X,Y) \le g(x,y)\} \cap \{Y>y\} ) = \text{?} = \int_{\xi = ...
0
votes
1answer
40 views

Deducing F-distribution PDF

Let $V\sim \chi^2(n)$ and $W\sim \chi^2(m)$ indep. r.v. I want to find the PDF for $X=\frac{V/n}{W/m}$. For that I define $h(v,w)=(v,v/n\cdot m/w)=(v,x)$. So, $h^{-1}(v,x)=(v,\frac{v \cdot ...
1
vote
1answer
24 views

Expected value of normal distributed variable

I need to calculate the expected value of a modified normal distributed variable but i'm struggling. So maybe someone can help me. Suppose we've got a normal distributed variable $X \sim ...
0
votes
1answer
16 views

How to determine the distribution of $U:=(X,Y,Z)$?

I've got a question concerning the distribution of a multi dimensional random variable. I know that $X$ and $Y$ and $Z$ are each normal distributed with certain expectations and variances. ...
1
vote
1answer
43 views

Coin toss with dynamic probabilities

So, I got a repeated experiment with two outcomes, i.e. a coin toss, but the probabilities might change every toss and are independent. Typically, they might come in sequences of the same ...
0
votes
2answers
44 views

If $X,Y$ ~$U(0,1)$ what is the distribution of $Z=0.5x^{2}+0.5y^{2}$?

I have some trouble with it.. the question is: $X,Y$ uniformly distributed $U(0,1)$ than $\frac{1}{2}(x^2+y^2) $~$exp(1)$... I am not even sure it is correct.. I know that if $X,Y$~$N(0,1)$ than it is ...