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

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1answer
167 views

maximum of exponentials

I am really having difficulties to prove the following: consider $X_1,\dots, X_n$ all exponentially distributed with rate $\lambda$ (i.e. $X_i \sim exp( 1/\lambda)$). Then argue that we can write ...
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1answer
657 views

Limit of Binomial distribution

In showing us that Binomial distribution: $$B_{N,p}(n) := \binom {N}{n} p^n(1-p)^{N-n}$$ tends to Poisson's: $$P_ \lambda (n) = \dfrac {\lambda ^n}{n!}e^{-\lambda}$$where I guess lambda should be ...
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1answer
40 views

Value of c in a density function

I've been trying to figure out how the answer was obtained with this integral and I'm completely stuck. Please help explain how was this integrated to get to the answer... For positive integer n, ...
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1answer
84 views

Conditional expectation $E[X|Y<y]$

Let $X:\Omega \to \mathbb{R}$ and $Y:\Omega \to \mathbb{R}$. Consider the joint pdf $f_{XY}(x,y)$ and univariate pdfs $f_X(x)$ and $f_Y(y)$. Is it true that $E[X|Y < y]$ equals: $$ \displaystyle ...
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1answer
245 views

Convergence in distribution - Gamma distribution

If we have a random variable defined as $Y_{n}=\displaystyle\frac{X_n-n\alpha}{n\alpha^2}$, where $X_n$ is $\operatorname{Gamma}(n,\alpha)$ distribution, how can I prove that $Y_n$ converges in ...
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2answers
632 views

Can Bhattacharyya distance be greater than one?

I have two vectors, say $P$ and $Q$. I want to find the statistical overlap between two given that $P$ is my reference which I have modeled after Normal distribution and I have parameters for it. $Q$ ...
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0answers
46 views

What is a topic I could easily collect data on that follows a poisson distribution?

So I have a project for my stat class and we have to form a hypothesis or question that I can do probabilistic modeling on. I really want to do a topic similar to a poisson distribution but I'm a ...
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0answers
68 views

What is known about $n$ independent random variables that yield a “converse” to uniform sample of a coordinate from a surface of an $n$-sphere?

It's well-known that to sample a coordinate $(Y_1,\ldots,Y_n)$ from a surface of an $n$-dimensional unit-radius sphere, it suffices to generate $n$ independent random variables $X_1,\ldots,X_n$ from ...
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1answer
335 views

Cumulative distribution function or density for Compound Poisson distribution

I have the Compound Poisson distribution $$ \xi = \sum_{n=1}^N X_n $$ where N has Poisson ($\lambda$) distribution and $X_i$ are independent and identically distributed and have normal distribution. ...
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0answers
129 views

Useful approximation of the pdf

Good day to everyone. In my research work I came out with a function, which looks like this (it is the pdf of some random variable): $$f(x,\rho,\psi)=\frac{2}{\pi }+\sqrt{\frac{2}{\pi }} ...
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1answer
88 views

Stochastic variable equals indicator function?

An exercise in my statistics & probability theory course goes as follows: $\Omega = [0,1], \mathcal{B} = \mathcal{B}([0,1]), P$ the Lebesgue measure on $[0,1]$. We have the sequence of ...
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1answer
273 views

Statistics and Probabilities- Distributions

A quality control engineer tests the quality of produced computers. Suppose that 5% of computers have defects and defects occur independently of each other. I need to find the probability that the ...
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1answer
62 views

Are the multiplications of i.i.d random variables , i.i.d?

If we know that $X_1$ and $X_2$ are i.i.d random variables, and $Z_1$ and $Z_2$ are also i.i.d random variables, can we say $X_1Z_1$ and $X_2Z_2$ are i.i.d random variables too? suppose that $X_1$ ...
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0answers
28 views

Suggest Power law distributions reference

What is everyone suggestion for a good Power Law Distribution reference in book or article form? Both theory and application is helpful (two separate books is fine). Thank you.
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0answers
78 views

About Strict Stationary of AR(1) Sequence

The usual Auto regressive process considers the time t from negative infinity and positive infinity, but what if we restrict our time to strict positive space, do we still have our stationary result? ...
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1answer
153 views

find the repartition function: $ F(x)=\int_{-\infty}^x f(t) \, dt$

Let $f$ be the function: $$f(x)=\begin{cases} k^2 xe^{-kx} & \text{if } x\geq0, \\0 & \text{if } x <0. \end{cases}$$ I have to prove that $f$ is a repartition density for every $k ...
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2answers
42 views

Obtain the cumulative distribution function of $X_1+X_2$

Suppose $X_1$ is a standard normal random variable. Define $$X_2=\begin{cases} -X_1, &\text{if} \,\, |X_1|<1 \\ \,\,\,\,X_1, & \text{otherwise}\end{cases}$$ Obtain the cumulative ...
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2answers
431 views

integrating using student t distribution

Evaluate the integral $\int_0^\infty\frac{1}{1+x^2}dx$ using the Student t distribution. I don't know where to start. I am assuming that I can't just do regular integration. I don't know how I am ...
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1answer
1k views

Independence between Uniform distribution and Exponential distribution question

I am trying to solve the following problem and I am having a great deal of difficulty in a number of areas. Help would be greatly appreciated! Let me state the problem first. If $X$ is uniformly ...
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1answer
314 views

Limits of Integration for marginal pdf

I just had a small question as something is bothering me. I am trying to find the marginal pdf of the following joint pdf: $f(x,y) = (1/8)(y^2 - x^2)e^{-y}$ where $-y \le x \le y$, $0 < y < ...
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1answer
170 views

Poisson Distribution Overlap

The arrival of clients in a store is modeled as a poison process with λ=3 clients per hour. What is the probability that 3 Clients arrive between 10:00AM-11:00 AM and 3 clients arrive between 10:30 ...
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1answer
189 views

Joint probability density

The question is taken from one of the past year papers. Please tell me if my answer needs any correction/improvement, thanks! The joint density function of $X$ and $Y$ is given by $f(x,y) = ...
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2answers
135 views

Generating a random number from a given distribution

I have a problem (a part of a Monte Carlo simulation) where I'm given the energy of an incoming particle, $\varepsilon$ and want to split this energy in two parts, randomly generating the fraction ...
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2answers
186 views

Cumulative Distribution of X/Y

Let X, Y be independent exponential variables with rates $\alpha$, and $\beta$. Find the c.d.f. of X/Y. So far, I let Z = X/Y. I can then show $f_Z(z) = \int_{-\infty}^{+\infty} |x|f_{X,Y}(x,xz) ...
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1answer
96 views

An almost-uniform distribution over the Naturals. Does it have a name?

I've just discovered this distribution, which is almost uniform over the naturals. Surely I'm not the first to discover it? Is it interesting? Does it have a name? $ \mathrm{P}(K = k) = \frac1k ...
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1answer
75 views

Marginal Probability Density: Integrand Values

I have a joint probability density function, $f(x,y)$. However, I have a constraint associated: $0< x < y < +\infty$. So, when I calculate the marginal probability densities, how do I ...
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1answer
34 views

How to compute future's (tomorrow's) distribution given today's?

Marital status can be defined as single, married, separated, or divorced. Today's distribution is: Single = 49% Married = 25% Separated = 15% Divorced = 11% I ...
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4answers
82 views

Distribution of a random variable

$X_1$, $X_2$, $X_3$ are independent random variables, each with an exponential distribution, but with means of $2.0, 5.0, 10.0$ respectively. Let $Y$= the smallest or minimum value of these three ...
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1answer
44 views

Distribution of means

Please help with a problem of practical application (explained by an example): Suppose there are 10 objects and 30 values (or scores). Suppose also there are 1000 students randomly selected from a ...
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1answer
37 views

If there are two geometric random variables $X_1$ and $X_2$, how to prove $E[X_1^2]E[X_2^2] \geq (E[X_{1}X_{2}])^2 $?

If there are two geometric random variables $X_1$ and $X_2$, how to prove $E[X_1^2]E[X_2^2] \geq(E[X_{1}X_{2}])^2 $? In addition, under what condition will the equality ...
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2answers
686 views

If $p_1 = 0.3$ and $p_2 = 0.4$, what is the probability that it will take Jay more than 12 hours to be successful on both jobs?

Jay has two jobs to do, one after the other. Each attempt at job $i$ takes one hour and is successful with probability $p_i$. If $p_1 = 0.3$ and $p_2 = 0.4$, what is the probability that it will take ...
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0answers
51 views

Finding a conditional probability from other probabilities

I am given the following probability distributions: $f_A$ = $\begin{bmatrix}0.2 & 0.8\ \end{bmatrix}$ $f_{B|A}$ = $\begin{bmatrix}0.4 & 0.6\\ 0.3 & 0.7 \end{bmatrix}$ $f_{C|A}$ = ...
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2answers
3k views

An ambulance problem involve sum of two independent uniform random variables

An ambulance travels back and forth at a constant speed along a road of length $L$. At a certain moment of time, an accident occurs at a point uniformly distributed on the road.[That is, the distance ...
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1answer
3k views

Find marginal density function from joint density function

If I have a joint density function for X and Y: $f_{X,Y}(x,y) = \begin{cases} \pi x \cos(\frac {\pi y} 2) & 0 \le x \le 1, 0 \le y \le 1 \\ 0 & \text{otherwise} \\ \end{cases}$ How do I ...
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0answers
63 views

How to read a probability distribution given as a matrix?

I'm currently reading up to try and complete an assignment but our lecture notes are very sparse and skip over the basics to instantly move onto solving random questions. Because of this I'm having ...
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0answers
68 views

Which probability distribution does my discrete variable approximate?

The discrete random variable X is a count variable of 45 Bernoulli trials and can take on values from 0 to 45. The mean is 2.34 and the variance is 18.39. The average probability of scoring 1 in a ...
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0answers
55 views

Is there any complete mathematical model for demand prediction?

Assuming that I have the history of the demand as a function of price $p \in [p_{min},p_{max}]$. I need a model better than the linear regression to predict the demand and its elasticity. The model ...
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1answer
101 views

Probability distribution of a sum of random variables

Suppose we have two independent random variables $X$ and $Y$. Their sum is $Z = X + Y$. How can I calculate the conditional distribution of $P(Z|X)$?
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0answers
48 views

What is the difference between these two questions?

Consider two independent random variables: X~U(0,1) and Y~U(0,2). Let Z = min(X,Y). b) Find F$_Z$(z) in terms of F$_X$(.) and F$_Y$(.). c) Eliminate F$_X$(.) and F$_Y$(.) to find F$_Z$(z). What is ...
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0answers
273 views

exponential and gamma distribution, survival functions

I have a homework problem, that I believe I can solve correctly, using the exponential distribution survival function. But, I think, I should also be able to solve it more easily using a gamma ...
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2answers
77 views

probability normal distribution

A model for the movement of a stock supposes that if the present price of the stock is s, then after one time period it will be either (1.012)s with probability 0.52, or (0.99)s with probability ...
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1answer
75 views

What is an Hypergeometric distribution where the last event must be a success?

I'm trying to find out the name of a distribution that is like negative binomial, only for finite population and without replacement. Or like Hypergeometric distribution where the last event has to be ...
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1answer
60 views

Inequality between two Random Walks

Let's consider two Random Walks, $$x^{(1)}_t = x_0 + \sum_{i=1}^{t}\xi^{(1)}_i,$$ $$x^{(2)}_t = x_0 + \sum_{i=1}^{t}\xi^{(2)}_i.$$ The random variables $\xi^{(1)}_i$ are i. i. d. They take values on ...
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1answer
519 views

Uniform sum distribution

I was wondering how to derive the probability density function for the sum of $n$ independent iid distributed random variables on the interval $[0,1]$. A formula for that is given on ...
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2answers
168 views

$E(X\mid X\lt x)$ with $X\sim\text{Exp}(a)$

$X\sim\text{Exp}(a)$. How do I calculate $E(X\mid X\lt x)$? Workings: \begin{align}E(X|X<x)&=\int_0^txf(x|x<x)dx\\ &=\frac{\int_0^txP(x,x<x)dx}{\int_0^tP(x<x)dx}\\ ...
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1answer
161 views

Memorylessness of the Exponential Distribution

Please help me solve the following question with two parts. $T$ is the time required to repair a machine. We have that $T$ is exponentially distributed with a mean of $\frac{1}{2}$ hours. For the ...
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1answer
231 views

Generating a random probability vector

Suppose that I want to generate a random probability vector $p = (p_1,\dots,p_d) \in [0,1]^d$, distributed uniformly over the simplex of probability vectors in $\mathbb{R}^d$. I would like to generate ...
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1answer
40 views

Binomial Cumulative Distribution probability

If you were to do a thing $x$ times, and 35% of the time it worked, what's the chance that after $x$ times it would have worked 240 or more times?
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1answer
152 views

successive doubling the stake until head appears

I consider the following gaming system: Start with 1 dollar and always bet on head (coin tossing). You always double your stake until the first head appears. Maximum rounds: $n$ I formulated it as a ...
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1answer
164 views

A problem on multinomial distribution

Suppose that an experiment can result in one of $r$ possible outcomes, the $i$th outcome having probability $p_i$, $i=1,\dots,r, \sum _{i=1} ^{r}p_i =1$. Let $X_i$ denote the number of times the ...