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

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1answer
14 views

Suppose that number of mistakes on a page is a Poisson RV and independent. From $n$ pages, find the expected number with no mistakes?

A textbook has $n$ pages. The number of mistakes on each page is a Poisson RV with parameter $\lambda$ and is independent of the number of mistakes on all other pages. What is the expected number of ...
0
votes
1answer
339 views

Mixture Gaussian distribution quantiles

Let $f_1(x), \dots, f_n(x)$ be Gaussian density functions with different parameters, and $w_1, \dots, w_n$ be real numbers that sum-up to unity. Now the function $g(x) = \sum_i w_i f_i(x)$ is also a ...
4
votes
1answer
79 views

Distribution of $\frac{X}{|Y|}$, where X and Y are standard normal r.v.'s

Let X and Y be independent standard normal random variables. What is the distribution of $\large \frac{X}{|Y|}$? Attempt: Let $\large U = \frac{X}{|Y|}$ and $ V = |Y|$. This transformation is not ...
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1answer
12 views

Conditional probability with four random variables

Assume A, B, C, and D are i.i.d random variables and k is a fixed constant. I want to find $\textbf{P}(A < B, C, D | D = k)$. How would I go about getting this, in terms of the cdf of these random ...
2
votes
1answer
92 views

A proof which results in Gamma (or Erlang) distribution-From Karlin & Taylor's “A First Course in Stochastic Processes”

The random variables X and Y have the following properties: X is positive, i.e., $P\{X > 0\} = 1$, with continuous density function $f_X(x)$, and $Y\mid X$ has a uniform distribution on $\{0,X\}$. ...
0
votes
1answer
28 views

Expected value of a function of a random variable [duplicate]

Let X be a random variable whose PDF is $f(x)$, and $g$ a function of random variable X. I want to prove that $$E[g(X)] = \int{g(x)f(x)dx} $$ I've perfectly understood it in discrete case and I ...
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0answers
11 views

Order statistics: does distribution of sum of two of them uniquely determine parent distribution?

Let $X_1, X_2, \ldots, X_n$ be a sequence of i.i.d. r.v. with bounded range (say, the interval [0,1]), with cdf $F$. Let $Y_1 \geq Y_2, \ldots, \geq Y_n$ be the corresponding order statistics. My ...
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0answers
24 views

Upper and lower bounds on probability in binomial distribution.

Suppose i have a random variable $X \sim \mathrm{Bin}(n,p)$ and some $1 \leq l \leq n$ can i obtain good upper and lower bounds on the probability that $$\mathbb{P}(X \geq l)?$$ After some research I ...
1
vote
1answer
320 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
14 views

How random are numbers from geometric distribution

If I will choose a value $r$ as a random number from uniform distribution, I can be sure, that this value is totally random - because each value is equiprobable. However, what if I will take a value ...
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0answers
23 views

PDF of product of two dependent r.v

I have a random variable q and another as the $min(X,q)$. $q$ and $X$ are continuously distributed r.v. I have the pdfs for both, but now I need to find the co-variance of $q$ with $min(X,q)$. ...
1
vote
2answers
43 views

Find the mean, variance and $f(21)$ of this if it is a p.g.f.

$$G(t) = t^{11}e^{9t-9}$$ Find the mean, variance and $f(21)$ Please help me with find $f(21)$ as easily as possible. For the mean and variance: $$G'(t) = 11t^{10}e^{9t-9}+9t^{11}e^{9t-9}$$ ...
4
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3answers
1k views

Derivation of the density function of product of two random variables

I am looking for distribution of product of two random variables. Best I could found so far was this formula from the relevant Wikipedia page: $$ f_Z(z) = \int_{-\infty}^{+\infty} \frac{1}{|x|} ...
0
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1answer
14 views

How to know the result of entropy function using uniform distribution set

In the entropy function here $H(s) = -\sum P(class=i|S)log_2{P(class=i|S)}$ I am trying to understand what is the domain of it's output for any input. I know that given a set where the frequency of ...
1
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0answers
191 views

Create the most 'stressful' tennis game ever!

Some games, such as tennis, use a complicated points system (point, game, set, match; with deuces and tie-breaks) for what would otherwise be an extremely simple and monotonous game. The main reason, ...
0
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1answer
20 views

Intro to probability chapter 4 ex 31

A group of 50 people are comparing their birthdays (as usual, assume their birthdays are independent, are not February 29, etc.). Find the expected number of pairs of people with the same birthday, ...
1
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3answers
485 views

How to prove uniform distribution of $m\oplus k$ if $k$ is uniformly distributed?

All values $m, k, c$ are $n$-bit strings. $\oplus$ stands for the bitwise modulo-2 addition. How to prove uniform distribution of $c=m\oplus k$ if $k$ is uniformly distributed? $m$ may be of any ...
1
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0answers
19 views

Total probability distribution of multiple random lotteries

My question: Imagine $d$ identical lotteries. Each individual lottery picks a cost $c_{i}$ between $0$ and $1$. Picking a costs occurs with probability distribution $f(c)$. The total cost of these ...
2
votes
1answer
36 views

$X \sim Rice(\nu,\sigma)$, what is the distirbution of $X^2$?

Let $X = |\nu e^{j\theta}+W|$, where $W \sim \mathcal{CN}(0,2\sigma^2)$, i.e. $X\sim Rice(\nu,\sigma)$, what is the distirbution of $X^2$? Note that X also can be writen in terms of real and imaginary ...
3
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2answers
51 views

Evolution of a discrete distribution of probability

I am designing a virtual card game and I defined an evolution of probabilities, but I don't have the knowledge on this matter to find out how they will evolve. I hope you help me here, with ...
0
votes
1answer
19 views

Given the distribution of $X$, whats the distribution of $cX$

Let's say $X \sim \chi_k^2(\lambda)$ with pdf $f_x(x)$ (i.e. noncentral chi-squared distribution). What can we say about the distribution of $Y = cX$ ? where $ c \in \mathbb{R}^+$ I know that $f_y(y) ...
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2answers
24 views

What is the best choice for $\mu$

A bottle is underfilled if it is less than $500$ml and overfilled if it is greater than $560$ml. Suppose the purchaser fines the supplier ${$}1$ per underfilled bottle. It costs the manufacturer ...
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0answers
11 views

Modifying a generator of random numbers from a trapezoidal distribution to include growth and decay rates

I've written a C# random number generator based on page 11 of this paper: http://pubs.usgs.gov/tm/04/c03/tm4-C3_final_508_files/tm4-C3_apdx1_v030813.pdf It works fine but I would like to modify it, ...
3
votes
0answers
35 views

Sufficient statistics and UMVUE for joint poisson, bernoulli

Given a pair $(X,Y)$ of r.v.s such that: $$X \sim \text{Poisson}(\lambda)\quad \text{and}\quad Y \sim B(\frac{\lambda}{1+\lambda})$$ with $X,Y$ independent, determine a one-dimensional ...
0
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0answers
10 views

What is the difference between the error wrt to the true distribution $\mathcal{D}$ and the empirical distribution $S$?

I was reading a paper and on page 4 they talk about the error of predictor with respect to the true distribution $\mathcal{D}$ and the empirical distribution $S$. In other words: $$Err_{ \mathcal{D}} ...
1
vote
3answers
58 views

How to prove $\sum_{i=1}^{n}\binom{n}{i}p^i(1-p)^{n-i}i = np$?

How to prove, when $p\in[0, 1]$, $$\sum_{i=1}^{n}\binom{n}{i}p^i(1-p)^{n-i}i = np$$ Is there a name for this formula?
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3answers
85 views

If the joint distribution is uniform, then the random variables are independent?

This is a problem that I am stuck at. If $X_1$ and $X_2$ are independent, it would be easier. But, the problem asks me the converse. For (i), I suspect that $X_1$ and $X_2$ are independent. But I ...
5
votes
2answers
909 views

Random number generation inside an interval based on cdf (Zipf and Exponential)

Consider for example the Exponential distribution with c.d.f. $F(x) = 1-e^{-\lambda x}$. $F^{-1}(x)$ would be inverse cdf (quantile function). If I generate y=F−1(x) with x uniformily distributed on ...
1
vote
2answers
34 views

Sums of independent random variables (more than two) [closed]

I read that the convolution of two iid random variables is $$(f * g) (z) = \int f(z-y) g(y) dy$$ What is the general formula for more than two RVs? For example, for three RVs.
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2answers
24 views

Expected Value and Variance of transformed Random variable

I am trying to find the expected value and variance of $Y_i=\ln(X_i)$ for $X$ is uniformly distributed between $1$ and $3$. I believe that $E(Y_i)=(\ln3)/2$ and $\operatorname{Var}(x)=(\ln3)^2/12$. ...
1
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2answers
28 views

Suppose $X$ and $Y$ are independent Poisson random variables. Find the conditional probability mass function $P(X=k\mid X+Y=m)$ [duplicate]

Suppose $X$ and $Y$ are independent Poisson random variables with parameters $\lambda$ and $\mu$, respectively. Find the conditional probability mass function $P(X=k\mid X+Y=n)$. Don't know how to ...
0
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0answers
12 views

How to choose assymetry for KL divergence?

I have two 2D probability distributions of eye movements of two different images. Suppose I call the first distribution of Image 1: $P$, and the second distribution of image 2: $Q$. Since ...
-1
votes
1answer
18 views

Suppose $X$ and $Y$ are independent Poisson random variables. Find the joint probability mass function $P(X=k, Y=m)$. [closed]

Suppose $X$ and $Y$ are independent Poisson random variables with parameters $\lambda$ and $\mu$, respectively. Find the joint probability mass function $P(X=k, Y=m)$. I know what a Poisson ...
2
votes
1answer
39 views

Product of Uniform random variable

Given three iid random variables, $X,Y,Z$ following a Uniform $(0,1)$ distribution. Then, how to find $\Pr(X>YZ)$? Recently, I was asked this question in an interview, but I chocked there. ...
1
vote
0answers
22 views

joint characteristic function of X and F(X)

X is a random variable. Its distribution function and characteristic function are $F_X$ and $\phi_X$, respectively. Then, we know, $F_X(X)$ follows uniform distribution. Let's say, $U=F_X(X)$. My ...
0
votes
1answer
385 views

How do you transform Gamma to Chi-squared distribution

Here is the question not sure how to turn a Gamma into a Chi-Squared: Suppose $X_1....X_n$ is a sample from the distribution Gamma($\alpha=3,\ \lambda=\theta$) with unknown $\theta > 0$. We wish ...
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0answers
13 views

Distribution of transformed poisson distribution

Let X follows Poisson distribution with parameter $\lambda\gt0$ and $Y=aX$, where $a\gt0$ is a constant. Q. What will be the $PMF$ of $Y$? Since by using the M.G.F we have ...
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0answers
15 views
4
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1answer
47 views

Computing Conditional Variance

I have been tasked with trying to solve a conditional variance. I have red and black pens with respective exponential probability parameters 2 and 4. I have 70% red pens and 30% black pens. What is ...
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0answers
4 views

Hierarchy-level profit distribution formula

I have a requirement and would like your expert opinion: I have a situation where user1 buys a product and then publishes an ad on his website about the sale of the product he purchased. Second user2 ...
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2answers
26 views

joint probability and conditional probability question

The number of workplace injuries, $N$, occuring in a factory on any given day is Poisson distributed with mean $\lambda$ . The parameter $\lambda$ is a random variable that is determined by the level ...
1
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1answer
20 views

Let $X~Bin(n, \lambda/n)$, $\lambda >0$. Show that for fixed $k \geq 0$, $P(X=k)\equiv \frac{e^-\lambda)\lambda^k}{k!}$

Let $X - Bin(n, \lambda/n)$, $\lambda >0$. By using approximation $(1-\frac{x}{n})^n\approx e^{-x}$. Show that for fixed $k\geq 0$, $P(X=k)\approx \frac{e^{-\lambda}\lambda^k}{k!}$ ...
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0answers
24 views

explanation : Example of Gaussian random process?

Can any one explain to me how to answer the question and what is the Gaussian random process in a simple way. I know how we find the C xx from R xx the rest of the answer I don't understand why all ...
2
votes
1answer
38 views

Marginal Densities

I just have a few questions about joint density and marginal density questions. Q1: Joint Distribution $f_1=2x+4y$ on triangle with vertices $(0,0), (0,1),(1,0)$. Sketch the region and compute ...
1
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0answers
25 views

conditional density of $X$ given $XY^2$

I was asked the following problem. Given that $X$ and $Y$ are random variables with joint density $f(x,y)$, find the conditional density of $X$ given $XY^2$. My thought was to first change variables ...
0
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0answers
28 views

Coalescent theory - Why are coalescent times independent?

I am reading from this book and I want to make sure I understand what is going on. What I get from the book Consider a population of $N$ individuals. The population size ($N$) is constant. select ...
1
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1answer
24 views

Convergence in Distribution and Exponential Function

There's a well known fact that if a sequence of real numbers, $\{x_{n}\}$ converges to $x$, then: \begin{equation*} \lim\limits_{n\rightarrow\infty}\left(1+\dfrac{x_{n}}{n}\right)^{n} ...
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0answers
30 views

independence copula diagonal

I'm reading Nelsen's Instruduction to copulas, and there is (probably very simple) excersice I cannot deal with. It says that if the diagonal section of the copula equals the diagonal of independence ...
1
vote
1answer
44 views

Convolution of Two Random Variables

I have been working a few hours on this particular problem. Please excuse my lack of formatting. This is the question: Let $X$ and $Y$ be random variables with density function $f(x) = 2x$ on $[0, ...
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1answer
66 views

Exponential Probability Question

A college buys 70% of dorm light bulbs from Company A with an exponential lifetime $f_A(x)~ exp(\lambda = 2)$. The other 30% come from company B have lifetime $f_B(x) ~exp(\lambda = 4)$. At the start ...