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

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17 views

Indicator variable syntax

Are these syntax equivalent? $$f(x,\lambda)=\lambda e^{-\lambda x}I_{(0,+\infty)}(x),\ \lambda > 0$$ $$f(x,\lambda) = \left \{ \begin{array}{cl} \lambda e^{-\lambda x} & x \gt 0 ...
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0answers
13 views

Determine eigenvalue distribution support

I am working on a project regarding random matrix spectra and I need some help with the following: let us assume we are looking at some particular family of NxN random matrices in the limit of N -> ...
3
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0answers
47 views

How to approximate the cumulative distribution function of the normal by a product of functions?

Suppose, there are $n$ vectors $\mathbf{X}_1$, $\mathbf{X}_2 \ldots \mathbf{X}_n$ of unequal lengths which can be combined to a new vector as $$ \mathbf{X} = \begin{bmatrix} \mathbf{X}_1 & \mathbf{...
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2answers
81 views

A fair die is rolled 100 times. Which of the following has a probability of at least 95%?

A fair die is rolled 100 times. Which of the following has a probability of at least 95%? $ $ 1.) Sum of the rolls is greater than 322 2.) Sum of the rolls is less than 392 3.) Number of rolls ...
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0answers
12 views

find out min/max of statistical distribution (GPA) from median, mode, count, size of elements? [closed]

I would like to find to the min/max of a distribution given the following. Was wondering if it is possible. You could think of them as GPA Number of elements in distribution: 93 Theoretical min of ...
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1answer
22 views

Moment generating function (MGF) of the ratio distribution $\displaystyle\frac{X}{Y}$

If we know the moment generating functions (MGFs) of the random variables $X$ and $Y$ to be $M_{X}(s)$ and $M_{Y}(s)$, respectively. The MGF of the sum $X+Y$ will $M_{X}(s) \cdot M_{Y}(s)$. So what ...
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0answers
38 views

Central Limit Theorem for gambling return ratio

Consider a single bet with odds $o$ and thereby implied probability $1/o$. Assume that the real probability $p$ is known. Let $I$ be the stake, and $y$ the return from the bet. Then, $\mathbb{E}(y) ...
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0answers
15 views

Queue depth to keep workers busy

I'm trying to find a probability of keeping w workers busy with a q queue depth feeding those w workers. When the queue has at least one item in it the item can be taken and the item was randomly ...
2
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1answer
26 views

Understanding the flat (uniform) Dirichlet distribution density over a simplex

This should be really straightforward from the formula, but somehow I'm having trouble understanding the density of a Dirichlet distribution with $\alpha = [1, 1, ... 1] \in R^k$, which is a uniform ...
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0answers
10 views

Finding $P(S<0)$ with standard Normal Cumulative Distribution function

I know I'm supposed to use the the Standard Normal Cumulative distribution function. But I can't seem to get everything I need. Let $X$ be a random variable with $P(X=-1)=P(X=0)=0.25$ and $P(X=1)=0.5$...
2
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1answer
182 views

Find the almost sure limit of $X_n/n$, where each random variable $X_n$ has a Poisson distribution with parameter $n$

$X_{n}$ independent and $X_n \sim \mathcal{P}(n) $ meaning that $X_{n}$ has Poisson distributions with parameter $n$. What is the $\lim\limits_{n\to \infty} \frac{X_{n}}{n}$ almost surely ? I ...
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1answer
27 views

Joint density function of $T_1,T_2$ and expectation of $E[T_1 ^2 +T_2 ^2 ]$

Given that $T_1,T_2$ are random variables representing the useful life (in hours) of two electrical appliance. The joint probability function of two variables distributed uniformly in the domain ...
2
votes
1answer
25 views

Confusion in Calculating Conditional Probability mass function

Question: If $X_1$ and $X_2$ are independent binomial random variables with respective parameters $(n_1,p)$ and $(n_2,p)$, calculate the conditional probability mass function of $...
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0answers
44 views

Integral evaluation - Gamma distribution

I have a sequence of independent random variables which are $\chi^2(1)$ distributed, $(X_i)_{i=1}^n$, $X_i\sim\chi^2(1)$. If I consider the sum $\frac{t}{n}\sum_{i=1}^n{X_i}$ this should be $\sim\text{...
2
votes
1answer
42 views

Distribution of $\lceil X \rceil - X$ where $X$ has an exponential distribution

Suppose $X$ is a random variable with exponential distribution of parameter $\lambda > 0$. That is, $X$ has density $f(x) = \lambda e^{-\lambda x} \mathcal{1}_{\mathbb{[0,\infty [}}$. The question ...
2
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2answers
38 views

How to calculate probability distribution of a function of two independent Poisson random variables?

I can't figure out how to determine the probability distribution function of $$aX + bY,$$ where $X$ and $Y$ are independent Poisson random variable. Basically, I want to check whether $aX+ bY$ ...
2
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1answer
36 views

Expected number of duplicates

Suppose I have $m$ bins and throw $n\ll m$ balls into the bins uniformly at random. (In my application $n\sim m/\log m.$) What is the expected number of duplicates? In other words, if there are $k_i$ ...
3
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1answer
28 views

Proving that a positive-integer valued random variable has the lack of memory property iff it has a geometric distribution.

Suppose that $X$ is a positive-integer valued random variable with the lack of memory property which states: Given that $X>n$, then $\mathbb{P}(X=n+k) = \mathbb{P}(X=k)$. Consider the case ...
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0answers
32 views

Moment Generating Function for $r$th central moment

When using moment generating functions, to find the $n$th raw moment ("$n$th moment about the origin"), you take the $n$th derivative of the MGF and evaluate at $t=0$. To find the $m$th central ...
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1answer
51 views

Expected value of an infinite sum of random variables

For k=1,2... let Xk be independent and identically-distributed random variables with E(Xk)= $\mu$ and V(Xk)= $\sigma^2$ and let N be independent of the Xk with mean $\lambda$ and variance $\lambda^2$. ...
7
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1answer
197 views

Is there any probability model for multi-stage motion of an object such as this.

I have this following case (please refer to attached pic below) where a particle is resting on the ground and it needs a minimum amount of force (Fmin) to reach from one level to the next level. But ...
2
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0answers
26 views

2D Cauchy Distribution Peak [closed]

Is the general form of a 2D Cauchy Peak, if A is the amplitude: $$\frac{A}{1+\frac{(x-x_0)^2}{\gamma_x^2}+\frac{(y-y_0)^2}{\gamma_y^2}}$$ $?$
4
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1answer
69 views

Calculate a limit $\lim_{n\to \infty} \frac{1}{n} \sum_{k=1}^n \Big\{\frac{k}{\sqrt{3}}\Big\} $

The problem is to calculate a limit $$ \lim_{n\to \infty} \frac{1}{n} \sum_{k=1}^n \Big\{\frac{k}{\sqrt{3}}\Big\} $$ where {$\cdot$} is a fractional part. I believe that this limit is equal to $\...
2
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2answers
53 views

operations on probability distributions

I've found that you can do certain arithmetic operations on random variables such as : multiply or divide two log-normal distributed variables add or divide two gamma distributed variables I've ...
2
votes
1answer
41 views

Probabilistic constraint implying deterministic constraint?

Suppose $X$ is an $N$-dimensional random variable $X := [X_1 \; X_2 \; \cdots \; X_N]$ such that all entries can either be 0 or 1 while satisfying the following: (i) $\mathbb{P}(X_i = 1) = p_i \; \; ,...
0
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0answers
20 views

Lorentzian Peak for Ellipse

I have the $(h,k)$ center coordinates, semi-major axis and semi-minor axis of an ellipse. I also have the height of the 2D Lorentzian peak, which is equivalent to the height of all the 1D Lorentzians (...
0
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0answers
23 views

Distribution of sum of weighted geometric random variables

Take $g_i$ to be a geometric random variable with parameter $1/2$, such that $$P(g_i = k) = \frac{1}{2^{k+1}}$$ for any integer $i$. I'm surprised at how much more difficult it is to evaluate this ...
3
votes
2answers
166 views

Calculating integral $\int_{0}^{\infty}x^2 \frac{f'(x)^2}{f(x)}dx$

This is a follow up question for this question: How can I calculate or simplify the following integral $$\int_{0}^{\infty}x^2 \frac{f'(x)^2}{f(x)}dx$$ If I know f(x) is a probability density ...
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0answers
15 views

Given probability generating function (PGF) for $X$, find PGF for $Y=2x+1$

PGF for $X$ is given by $G_{X}(t)=t^2e^{t-1}$, where $t\in\mathbb{R}$. Find PGF for $Y=2X+1$. I started it doing this: $G_Y(t)=\mathbb{E}t^{2X+1}=t\cdot\mathbb{E}t^{2X}=t\sum_{k=0}^{\infty} t^{2k}\...
0
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1answer
43 views

Distribution of a exponetial Random Variable

i have a stopping time $T$ of an Poisson Process $N$ with rate $\lambda$. Somehow this stopping time is exponential distributed. So we have $ T \sim exp(\lambda)$. I want to know the distribution of ...
1
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2answers
55 views

Proofs related to chi-squared distribution for k degrees of freedom

I was reading a proof related to chi-squared distribution for k degrees of freedom from wiki. https://en.wikipedia.org/wiki/Proofs_related_to_chi-squared_distribution I think I might understand the ...
1
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0answers
9 views

Joint distribution of sum and summand

Let $Z_1$ and $Z_2$ be independent random variables with known distributions $F(.;\theta_1)$ and $F(.;\theta_2)$ of the same convolution closed family. Then $Y = Z_1 + Z_2$ has distribution $F(.;\...
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0answers
12 views

Is this probability density function (PDF) estimation, $\hat f_{X:\mathcal{X},\theta^*}$, optimum?

Questions: Q1: Is PDF $\hat f_{X:\mathcal{X},\theta^*}$ an optimal estimation of PDF $f_X$? Q2: What are the conditions that, if met, $\hat f_{X:\mathcal{X},\theta^*}$ is optimum? E.g. would it ...
1
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1answer
30 views

Knowing that the probability to drill success (produced) exploratory hydrocarbon well is $0.2$ by certain company

Knowing that the probability to drill success (produced) exploratory hydrocarbon well is $0.2$ by certain company, this company has drilled four success exploratory wells in different areas on the ...
-1
votes
1answer
67 views

Let $X \sim\operatorname{unif} (1,2)$. Find the distribution of $ Y=X+2/X $

If $ X $ follows the uniform distribution in $ (1,2) $ what is the distribution of $ Y= X + \frac{2}{X} $ ? I thought that $ P( X + 2/X <=y )$ => $ P(X^2-2xy +2 <=0)$ , where y is at $(2\sqrt{...
0
votes
0answers
29 views

Probability distribution of $\omega'(n)$. [duplicate]

$\omega(n)$ is the number of distinct prime factors of $n$ and $\omega'(n)$ is the number of distinct prime factors of $n$ with multiplicity. For example if $p,q$ are prime numbers then $\omega(p^2q)=...
1
vote
1answer
55 views

Show that $\mathbb{E}(X-c)^2$ is minimum when $c = \mathbb{E}(X)$

Suppose that the random variable X has the cumulative density function F(x). Show that the expected value of the random variable $(X-c)^2$ is minimum if c equals the expected value of X. I know that ...
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0answers
13 views

Where did I make a mistake in this transformation of random variable?

The arctangent of a standard Cauchy random variable $Z\sim\text{Cauchy}(0,1)$ is uniformly distributed in $[-\frac{\pi}{2},\frac{\pi}{2}]$. The proof is straightforward: $$P(\arctan(Z)\leq t)=P(Z\...
1
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1answer
18 views

How to calculate the central moment giving function of a distribution

Is there a function which gives the central moments instead of just moments of a distribution and if so how to calculate this function for a distribution e.g. the normal distribution.
4
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1answer
85 views

PDF of the difference between two independent beta random variables

I am having trouble deriving the distribution of the difference of two beta random variables and would like some help verifying the steps I have taken. In particular calculating the bounds. Say I ...
0
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0answers
30 views

Application Strong Markov Property

I am considering a random walk $S_n$ on a state space $\mathbb{Z}^d$. I want to show that $E_x\left[\sum_{n=0}^{\tau_A-1}{1_{\{S_n=y\}}}\right]=\frac{1}{P_x[\tau_A<\tau_y]}$, where $\tau_A=\inf\{n\...
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votes
1answer
16 views

Convergence of time translated iid process

Asume we have a sequence of i.i.d. rv $(Y_n)_{n\geq 0}$, with finite expectation. If $\sqrt{n}^{-1}Y_n\rightarrow 0$ almost sureley, can one conclude, that $\sqrt{n}^{-1}Y_{n+m}\rightarrow 0$ almost ...
0
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0answers
20 views

Cumulative bivariate normal

How do I calculate the cumulative probability distribution function for a bivariate normal distribution with conditions $P( x>a , y>b)$? Is there any method to solve $$P(x>a,y>b)\\\int_{...
1
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0answers
33 views

Independent random variable in the limit (need extension?)

Given a sequence of rv $(X_n)$ on $(\Omega,\mathcal{F},P)$ with values on $(E,\mathcal{E})$ where $\mathcal{F}=\bigcup \mathcal{F}_n$ and $\mathcal{F}_n=\sigma(X_s:s\leq n)$. Lets say there is a ...
0
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0answers
9 views

Bounds for transition density and its derivative

Suppose the process $X_t$ has a transition density $p(t,x,y)$, which is continuously differentiable w.r.t $y$. In my proof, I use the following properties of $p$ and $p_y$: There exist functions $\...
0
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1answer
108 views

Average number of terms required for a sum of exponential variables to reach a specific limit

I have a sum $\sum_{i=1}^\infty Y_i$ where $Y_i=AX_i+a$ if $X_i>X_{lower}$ and $Y_i=BX_i+a$ if $X_i<X_{lower}$. Here $X_{lower}, A, a, B$ are positive constants and all $X_i$'s are i.i.d ...
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0answers
18 views

Writing the complementary cumulative distribution as an expectation over indicator functions

Here is a neat little relation and I am wondering if/how it generalizes. The complementary cumulative distribution functions of a random variable, $X>0$, with density, $\rho(x)$, is $$C(x)=\int_x^\...
0
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1answer
23 views

Joint Probability Distribution function

I have two random variables $\alpha(x)$ and $\beta(x)$, and they are correlated. $\alpha(x)$ obeys log-normal distribution while $\beta(x)$ obeys normal distribution. How do I construct a joint ...
3
votes
2answers
52 views

Determine whether a random binary sequence was generated by human or natural process

Given a binary sequence, how can I calculate the quality of the randomness? Following the discovery that Humans cannot consciously generate random numbers sequences, I came across an interesting ...