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

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3
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1answer
73 views

Higher mean implies the running sum is more likely to be greater than 0?

For simplicity, consider the sample space {-2,-1,0,1,2}. We have two distributions, say, $f$ and $g$ that are defined on that sample space. Let $x_i$ and $y_i$, $i = 1,2,\dots$, be the i.i.d. random ...
1
vote
1answer
47 views

Computing P-value

In a book, from a sample they derived Mantel-Haenszel chi-square statistic $$\chi_1^2=1.41$$ And it is written that : this $\chi_1^2=1.41$ is associated with a one-sided P-value between $0.10$ and ...
1
vote
1answer
116 views

Probability density functions of multiple random variables

Problem: Two birds have landed on a power line that spans the 100' distance between utility poles. a) What is the average distance between the birds? b) The line runs north and south. Another bird ...
1
vote
1answer
47 views

How to compute the integrals in inverse formula?

I have following characteristic function for certain random variable X: $$\Phi (t) = \frac{\beta_1\beta_2}{\eta_1}\frac{\eta_1 - it}{(\beta_1 - it)(\beta_2 - it)}$$ where $\eta_1 > 0, \quad\beta_1 ...
7
votes
1answer
93 views

Random walk on cubic lattice

Suppose at every point of the cubic grid in n dimensions is a particle, and at every timestep every particle moves at random to one of its 2n neighbours. As time goes to infinity, what is the ...
0
votes
1answer
258 views

Joint probability density function wherein two random variables are uniformly distributed on a quarter-circle

Random variables, X and Y, are uniformly distributed in the quarter circle, with center at the origin and a radius of one in the first quadrant of the x,y plane. Please find the joint PDF of X and ...
0
votes
1answer
69 views

Variance of sum of multiplication of independent random variables

Suppose that we have $Z=\sum_{i=1}^n (a_i+b_iX_i)(c_i+d_iY_i)$. Where $a_i,b_i,c_i$ and $d_i$ are real numbers and $X_i$s and $Y_i$s are all independent random variables. How can I find the variance ...
0
votes
1answer
215 views

Finding a joint probability density function given marginal probability density functions

X and Y are independent random variables with the following PDFs: $f_{x}(x) = \begin{cases} (1/3)e^{-x/3}, & x \geq\text{ 0} \\ 0, & \text{otherwise} \end{cases}$ $f_{y}(y) = \begin{cases} ...
1
vote
0answers
46 views

Is the plane created by ($\int f_0^u f_1^{1-u},\int f_1^u f_0^{1-u})$ continuous?

$f_0$ and $f_1$ are two continuous density functions on $\mathbb{R}$. I wonder if $$(x,y):=\Bigg(\int f_0^u f_1^{1-u},\int f_1^u f_0^{1-u}\Bigg)$$ for all $f_0$ and $f_1$ is complete for some ...
1
vote
0answers
115 views

Occupancy distribution bounds for $k$ balls in $m$ bins

Suppose we throw $k$ (distinct) balls into $m$ (distinct) bins, and let $B$ count the number of non-empty bins. I am interested in lower bounds on $B$. More precisely, I wish to bound from above the ...
1
vote
1answer
359 views

Distribution of sum of multiplication of i.i.d. exponential random variables.

I have two questions: A) Suppose that we have $Z=c\Sigma (X_i-a)(Y_i-b) $ where $X_i$s and $Y_i $s are independent exponential random variables with means equal to $\mu_{X}$ and $\mu_{Y}$ (for ...
1
vote
1answer
116 views

Relation between two distributions expressed in terms of their CDFs

Not great at stats, and having trouble wrapping my mind around this. Would love an explanation, not overly detailed, in plain english of what these transformations mean. The bias correction ...
0
votes
1answer
29 views

Finding the probability of $R_1$

Consider a binary communication system that consists of a transmitter, a receiver and a Chanel that transfers bits from the transmitter to the receiver. The nature of the channel is such that it ...
1
vote
2answers
87 views

The sum $Y$ of independent Bernoulli variables with Poissonian upper limit $N$ is independent of $N-Y$

The random variables are $N,X_1,X_2,..$ are independent,$N \in po(\lambda)$ and $X_k \in Be(1/2)$, $k \geq 1$. Set $Y_1 = \sum\limits_{k=1}^{N}X_k $ and $Y_2 = N - Y_1$. Here $Y_1 = 0$ for ...
0
votes
0answers
61 views

Conditional distribution for sum of random variables

Let $X$ be some discrete random variable, i.e. $\mathbb P(X=x_j)=p_j$ for $x_j\in\mathbb R$ and $\sum_{j=1}^J p_j=1$. Furthermore, let $L_1,\ldots, L_n$ be a list of be random variables of which we ...
2
votes
1answer
151 views

How to find the CDF of the sum of independent uniformly distributed random variables?

$X,Y$ are independent random variables with uniform distribution on $[0,1]$, and let the random variable $Z=X+Y$. The density of $Z$ is: $$f_{X+Y}(z)=\int_0^z f_X(x)f_Y(z-x)dx$$ What is the formula ...
1
vote
1answer
42 views

cdf is $F_X(x) = 1-(1-x)^k$, is that a “famous” distribution?

The question is the following. I found that the cdf of X is $F_X(x) = 1-(1-x)^k$, where $k$ is a parameter. I was wondering if that is some famous distribution (like the one that has a name, for ex., ...
1
vote
0answers
259 views

Approximation for the convolution of normal and lognormal distributions

$$X \sim \ln\mathcal{N}(\mu_X,\,\sigma_X)$$ $$Y \sim \mathcal{N}(0,\,1)$$ $$Z = X + Y$$ I want to find the probability density functions and cumulative distribution functions of $Z$. As the below is ...
2
votes
2answers
36 views

Product of binomial and uniformly distributed variables

I've got a question that I really think should be quite simple to answer, but I just can't see what I'm missing. We have the random variables $X \sim R(0,1)$ and $Z\sim b(1,1/2)$. I want to determine ...
3
votes
1answer
248 views

Expectation of first and second order statistics in a random distribution

Let $E(f_{i}^{n})$ and $E(s_{i}^{n})$ denote the expected first and second order statistics for $n$ draws from the distribution $V_i$ .i.e set $X_{i}^{n}=\{x^1,.....,x^n | x^j \sim V_i \}$ and let ...
0
votes
1answer
50 views

Calculating expectation including score function

I can't come up with an efficient solution of the following problem. Let $(X, Y)$ be a 2-dimensional random variable vector which follows the 2-dimensional Gaussian distribution, and its ...
1
vote
0answers
157 views

how to calculate remaining waiting time in exponential distribution?

ABC corp conducted a study of service times at the drive-up window of fast-food restaurants. The average time between placing an order and receiving the order at restaurant is 2.45 minutes. Assume ...
0
votes
1answer
52 views

How to find Reliability of a rectangular distribution function?

Assume that the failing of a device is equally probable within an interval [a,b] such that the fault density is: f(x) = {1/b-a if a<= t <= b ...
0
votes
2answers
95 views

What does “Choose N ~ Poisson(ξ), Choose θ ~ Dir ( α )” mean in the context of Latent Dirichlet Allocation

I'm reading http://machinelearning.wustl.edu/mlpapers/paper_files/BleiNJ03.pdf and trying to understand the notation and concepts behind LDA, in order to implement it myself. I've followed some ...
0
votes
1answer
38 views

limiting joint distribution

Let $X_n\xrightarrow[d]{}N(0,\sigma^2_x)$ and $Y_n\xrightarrow[d]{}N(0,\sigma^2_y)$. $X_n, Y_n$ are not independent. Can I say that $\left( \begin{array} {} X_n \\ Y_n \end{array} ...
1
vote
2answers
74 views

Conditional Expectation Discrete and Continuous

Find $E[X]$ and $Var[X]$ So for the expectation so far I got that: $$E[X] = E[X|N=n]P(N=n) = \large\frac{n+1}{\lambda} \frac{\lambda^{n}}{n!}e^{-\lambda}$$ but for conditioning on both a discrete ...
2
votes
1answer
38 views

Convergence , conditional distribution

here my short question. I saw the following conditional distribution, which converges, in a book: $\lim_{s \to \infty}P\left(\frac{X-f(s)}{g(s)}\leq x\mid X>s\right)=G(x)\ \forall x$ in the set ...
1
vote
0answers
92 views

Implementing the Delta method to assess the confidence and prediction intervals

I want to calculate the table of confidence and prediction intervals for a custom Cumulative Distribution Function or CDF, and I am following the forums and articles aid. My major cuestions that I ...
1
vote
0answers
41 views

Infinite fourth moment and maximum entropy

Alright, I expect this is a silly question, but I don't actually know, so. Suppose there is some random variable that's distributed on the reals, and all I know about the distribution is its mean ...
1
vote
1answer
29 views

Relation between the distribution functions of random variables $Y$ and $-Y$

I'm having trouble understanding a certain property of CDFs for negative random variables. Let $Y$ be an exponential random variable and let $f_y, F_Y$ denote the PDF and CDF respectively. My book ...
0
votes
1answer
93 views

Hypothesis Testing, P-value, T-test Statistic, Confidence Interval

I am writing a report for my class project. I am taking statistics and I am REALLY panicking with the results I have in my report. I do not think my calculations for t-test statistic or confidence ...
0
votes
1answer
34 views

Proving Continuity and equivalence

I have posted ths on the Quant Finance page as it is part of a QF problem but realised I may get a swifter response here! Iam working on a problem where I have successfully reduced a version of Black ...
0
votes
1answer
31 views

Combining biased coin flips

Suppose one has a biased coin $C_1$ with probability $p$ of landing heads and $(1-p)$ prob. of landing tails. If one wants to sample a coin $C_2$ with $p^2$ probability, one can flip the coin $C_1$ ...
1
vote
1answer
172 views

Determine the accuracy of Poisson approximation to birthday problem

I'm currently doing an exploration of the Birthday Problem, and noticed that the formula given to calculate the probability for $m$ people in a room is: $$1-\frac{365!}{365^m (365-m)!}$$ And this ...
1
vote
1answer
21 views

Convergence in distribution for changing domains.

I am trying to consider whether this is possible and/or reasonable: Let $X_n:\Delta_n \to \mathbb{R}$ be a sequence of random variables, defined over a unique space $\Delta_n \subseteq \Omega$ for ...
4
votes
0answers
72 views

Probability and sums of prime factors

Of the first $N$ natural numbers, we select two different numbers at random. We'll call the greater one $A$ and the lesser one $B$. What is the probability $P$ that the sum of $A$'s prime factors is ...
2
votes
1answer
43 views

$X$ ~ $\Gamma(s,\lambda)$. Using $M_X(t)$ find the following…

a) $E(X) =$ ? b) $E(X^2) = $ ? c) $Var(X) = $ ? My thoughts: I know that moment-generating function for $\Gamma(k,\theta) = ( 1 - t\theta)^{-k}$ for $t < \frac{1}{\theta}$. I also know that ...
0
votes
1answer
60 views

Mixing continuous and discrete distributions

I'm wondering how, if it is at all possible, to write the p.d.f. for the following random variable. Given RVs $X_1$ and $X_2$ distributed according to some joint distribution having known density ...
2
votes
1answer
60 views

Deriving a lower bound for a probability involving a random variable $X$ with the Gamma distribution.

Question Let $X$ have the $Gamma(\alpha, \beta)$ density. I.e. $$f_X(x) = \frac{1}{\gamma(\alpha)\beta^\alpha}x^{\alpha-1}e^{-\frac{x}{\beta}}$$ when $x >0$ and $0$ elsewhere. The moment ...
0
votes
2answers
37 views

Probability density use for biased outcome

I am not a mathematics pro so do not mind if this is dumb let us suppose I have a method for generating random real values between 0 and 1 . All the values between 0 and 1 are equally likely to be ...
1
vote
1answer
38 views

Probability density function of two uniformly distributed stochastic variables

I'm currently stuck on an exercise involving two independent stochastic variables X and Y. Both X and Y ~ U(0,1) (uniform distribution) The goal of the exercise is to calculate the probability ...
0
votes
1answer
48 views

Simple Expected value of MLE

Let $X_1,..., X_n$ be iid $Exp(\lambda)$. The MLE for $\lambda$ is $\hat{\lambda}=\frac{1}{\bar{X}}$, where $\bar{X}=1/n \sum^n_{i=1}X_i$ How can I conclude that $E(\hat{\lambda}) = n\lambda/(n-1)$? ...
0
votes
1answer
40 views

Distributions of local times of a single excursion of 1D random walk

Consider Simple Random Walk in one dimensions, starting from $x \in \mathbb{Z}^+$. The walker jumps to the right with probability $p$ and to the left with probability $1-p$. Assume $p \leq ...
0
votes
2answers
28 views

Finding out the probability distribution of numbers from -3 to 3

I am really able to solve problems with the help of all the volunteers here. a big thanks to every one.. Please explain this problem.. A random variable 'X' takes the values -3,-2,-1,0,1,2,3. ...
1
vote
1answer
28 views

Histogram with different sample probabilities

Assume we are given a list of samples $L_1,L_2,\ldots,L_n$ of some random variable $L$. By classing them into bins we can easily create a standard histogram. But now suppose that we associate a ...
1
vote
0answers
51 views

Working with the sum of two independent random variables, and estimating a parameter

A network source sends a sequence of zeros and ones, $X_1, X_2, ...$ with $X_i$(iid) Bernoulli with $p = P(X_i = 1), 0 < p < 1$. Due to disturbances the received sequence is $Y_1, Y_2, ...$ ...
3
votes
1answer
35 views

Is there a name for the probability distribution with the form $p(x) = a \,x^2\, \exp( -b\,x^2 )$

There is a probability distribution: $$p(x) = a \,x^2\, \exp( -b\,x^2 ), \quad a,b>0,\ x \in ( -\infty,\,\infty ) $$ I wonder which probability distribution is it?
1
vote
1answer
58 views

$E(X_i \cdot I(X_i>\theta)$ expected value of when X is greater than the median.

Let $X_1, ..., X_n$ be iid with a distribution F. Let $\theta$ be the median of F. What is the value of $E(X_i \cdot I(X_j>\theta))$? If $i\neq j$, then $E(X_i \cdot I(X_j>\theta))= 1/2 \cdot ...
2
votes
1answer
62 views

Finding conditional distribution

Let $X$ and $Y$ be independent $Exp(1)$-distributed random variables. Find the conditional distribution of $X$ given that $X + Y = c$ ($c$ is a positive constant). this is my idea: $$f_{X \mid ...
0
votes
2answers
162 views

Laplace transform of noncentral chi-square distribution

I'm interested in non central chi-square distribution. More specifically, i want to derive the laplace transform of noncentral chi-sqruae disribution or density function. Let me know whether it ...